PreCalculus Full Course For Beginners

all right so we are going to begin with some review for calculus um we're going to do a number of videos on pre-calculus review starting with a quick review of the set of real numbers okay so this notation this sort of bold-faced r you've probably seen before to you know this set of real numbers and you've probably seen in high school some discussion of different ways that you can interpret the real numbers right so one of the interpretations that you might have seen is you know well okay so first of all your your natural numbers are in there right your natural numbers your integers so you include the the negatives those are also in there all right your rational numbers those are in there and by the way these sets they also have they also have symbols just like r right the natural numbers the integers are typically denoted with a z rational numbers r is already taken so we use q for quotient okay so we have we have all of these and but that's not everything right um so the ratio that we know that they're irrational numbers like pi and e and the square root of 2 right so you throw in all the irrational numbers as well and and so basically we can think of you know you start with all these numbers and then you also sort of throw in you know any sort of number that you can write with a decimal expansion right so it might be like a natural or number or an integer where there is nothing after the decimal place it might be a rational number where you have either a terminating or a repeating decimal expansion but it might be something like an irrational number like square root of two or pi where the decimal expansion goes on forever never terminates never repeats this is one way that people will think about rational numbers is just in terms of of decimal expansions another interpretation that you've probably seen is this sort of idea of a number line right so we think of the real numbers as this you know the set of all points on this on this number line that goes on forever in either direction somewhere in the middle is zero with positive numbers on this side negative numbers on this side right and we think of this as the sort of continuous line there's no gaps or breaks or anything in it and every point on the line gives you a real number okay so that's fine that's one way to think about it the way that mathematicians actually think about the real numbers is we usually think of real numbers in terms of properties okay and i'm not going to list all the properties but there are quite a few of them there are the algebraic properties algebraic properties so these are specifying all the rules for how addition and multiplication behave right so so these are all the rules like you know the commutative property which says that a plus b is the same thing as b plus a right the distributive property so as an example we would have the fact that if you do a times b plus c that's the same thing as a b plus ac right that's known as the distributive property it's key to doing things like factoring and multiplying out polynomials right um the foil rule that you probably know comes from this distributive property so there are all these algebraic properties telling you how basically you know how to manipulate real numbers how to solve equations that's the algebraic side of things but there are also order properties now the the ordering has to do with this number line picture right real numbers are ordered from left to right right there's this idea of increasing order and there's sort of built in here there's sort of a notion of of size this idea that we can talk about one real number being bigger than another right and in fact um mathematicians would say that the real numbers are totally ordered right so given any two real numbers you can decide which one is bigger than the other right um given any three you can put them in order right you know small medium big right so you can any set of real numbers you can always put them in order right there there isn't anything kind of different tracks that things might go on there's just one single straight line right increasing size so we have all those order properties which come up when you're trying to let's say solve inequalities things like that so this is going to pop up again later so we won't we won't spell everything out now the the last one is i think the the hardest one it's kind of the key property of the real numbers because um the rational numbers for example have all the same algebraic properties that the real numbers have they have all the same order properties that the real numbers have okay but they don't have this continuum property and there are a number of different ways of of saying what the continuum property is they all of them kind of go beyond what we would do in a first calculus course though and they get more into what you might see in an upper division analysis course so one one way to think about what the continuum property means is that it kind of gets back to this this decimal idea so every um possible decimal expansion produces a real number the the precise statement of this would be would be stated in terms of sequences in series something you probably won't see until your second or third course in calculus but we know that this is not true for rational numbers right so there are lots of decimal expansions that don't correspond to rational numbers because if the expansion doesn't terminate and it doesn't repeat you don't have a rational number another way to phrase this continuum property is is there's also something you could put it in terms of what are called least upper bounds but that's not something that we want to get into okay so there's another way to describe it in terms of least upper bounds so these are all things to keep in mind when you're studying the real numbers and maybe the main thing to keep in mind is that truly understanding the real numbers actually having a definition of what it means for something to be a real number is beyond what we do in a calculus course despite the fact that everything we do in calculus depends on the real number system right the the playground for calculus is is the set of real numbers and yet we don't actually have a solid definition of what a real number is and the reason is that the real numbers are actually pretty hard to define and it's not something that we're going to try to do in this course right once you've got some comfort you know with functions with limits um with with calculus then you're in a position where you can actually start talking about what a real number is how would you define one what are the properties it turns out that this is actually a fairly advanced topic so don't worry if you never quite feel like you know exactly what a real number is because that's something that will come later on in your mathematical career okay so i know what you're thinking these videos are for calculus right what do we do talk about order of operations this is basic stuff this is well maybe not elementary school maybe middle school but despite the fact that this is basic material you're going to be wrestling with derivatives and integrals and and you know understanding the meaning of continuity you still have to do arithmetic you still have to be able to add up numbers you still have to be able to solve equations these things are still going to come up order of operations messes people up more often than you'd think right so people have their you know their various acronyms that they like for remembering this one of them you've probably seen is this head mass or bed mass depending on whether you call these things brackets or parentheses we'll call them parentheses so these rules say that well parentheses come first okay simple enough or maybe not so simple because you've got to watch out for things like nested parentheses you might have parentheses within parentheses within parentheses so you know you got to pay some attention there okay the e there is for exponents all right you sort of basic principle here is you should start with the most complicated operations and work your way down in the order that you carry them out right so exponents are are more complicated than say multiplication or division so those come next okay and last of all the a and the s is addition and subtraction fair enough um but but even spelling it out like that things are not quite as simple as it might seem um let's let's do a quick example let's say i give you something like 2 plus 3 to the 4 times 6 minus 8 divided by 3 times 5. um okay so there's there's an expression you can write down it involves you know well there's no parentheses yet but it involves all the other arithmetic operations and so i guess if you're going to carry things out in in order then this this exponent that should be the thing that you do first now there's there's one of the things that trips people up d then m does that mean we should do division before multiplication is this second well some people will say yeah do division first other people will say you know division and multiplication they kind of come as a group and you should just kind of carry things out working from left to right in the order that you see them there's actually no agreement on that some people will say one some will say the other you can find calculators that are programmed in one or the other of those two ways and and so you really should not write down an expression like this with an ambiguity like this you know eight divided by three times five which one do you do first probably you should put some brackets in here right put some parentheses in then it's clear so if i do this let's say like that now i understand what i'm doing right so i should do the 3 to the fourth what 81 i should multiply by the six right so i'm gonna in my next step i'm gonna have an 81 times six there two uh here i'm going to do the 8 divided by 3 well i mean it's not a whole number i guess we just have eight thirds right times 5. next we do those two multiplications then we do the addition and the subtraction that we get our answer right i'm not going to actually work out the number you can do that on your calculator if you want so you've got to be careful with things like that as much as this seems really basic there are a few things that tend to trip people up one of the things that trip people up is negative so be careful of negatives right so for example let's say i'm doing something like three minus two times one minus four okay a lot of people will say okay that's equal to three and they know okay um i guess if you're doing order of operations i suppose you should do the one minus four first right so really what we should do is we say three minus 2 times minus 3 then we remember oh negative times a negative that's positive right so 2 times 3 6 so 3 plus 6 and we get our 9. but more often than not what people are going to do is they're going to distribute the 2 right they're going to use the distributive property and they're going to say oh minus 2 and then they're going to say minus eight okay so i'll write it like that that's not right is it no three minus two minus eight that's not going to give me nine right and the reason it's wrong is that when you distribute that 2 right it's minus 2 times 1 and then minus 2 times minus 4. a lot of people forget that there's actually two negatives there and that should be a positive right that's an easy one to forget another one that's easy to forget is you have this notion of implied parentheses okay so you might have something like this 3 minus 2 divided by 5 plus 1. all right so if you're if you're doing something like that you've got to remember that hey actually you got to do the 3 minus 2 first you got to do the 5 plus 1 first right really what you have here is is 1 over over 6. all right um so where people tend to mess this up is there's a lot of people and i think this is kind of a keyboarding thing right you get into the habit of writing fractions and so instead of writing a fraction like this you write your fraction like that that saves a little space right fits on one line instead of two um but then what happens is you're doing a problem maybe you're doing like an online homework problem you're entering your answer into the computer you wanted to check it's telling you you're wrong and it's telling you you're wrong because you know you typed in something like this you typed in three minus two divide five plus one that's what you typed into the keyboard and you assumed that somehow the computer was gonna know that it should do this subtraction first and it should do the addition first and then it should do the division right computer doesn't know that computers are stupid computers are going to do exactly what you tell them to do and they're going to use the order of operations so you have to explicitly tell the computer by putting in the brackets that it needs to do those operations first then do the division or you'll get the wrong answer you've got to be careful about these things okay next step we're going to talk interval notation okay so intervals are kind of standard subsets of the real numbers and they pop up all over the place in calculus right maybe you're writing an interval because it's the solution to an inequality maybe you're writing enter an interval because it's the domain of a function right probably the most common scenario where you're going to be writing down intervals is you're getting towards the material on let's say curve sketching right you're looking at how derivatives shape the graph and you want to know things like where is the derivative increasing where or sorry where's the derivative positive where is it negative that tells you where a function is increasing or decreasing right and the way you specify this is is using intervals so you'll give an interval say well it's positive on this interval it's negative on this other interval right so basically you break the real number line into these pieces each of those pieces is going to be an interval and on those intervals something interesting is happening to your function okay so there are a number of different types of intervals and there's different notation used for each and that might be some of what throws people off so the basic types are the open interval which is written like this and the closed interval and instead of using parentheses we use square brackets okay so what are these numbers right so in in all of these here assume that a is a smaller number than b right the these are referred to as the left endpoint and the right endpoint of the interval the left endpoint is always smaller than the right endpoint right because we always kind of increase as we move from left to right in the real numbers so an open interval is as an inequality if you like we could write like this as a set so it would be the set of all real numbers x that satisfy the following inequality x has to be bigger than a but smaller than b okay for the close interval it's almost the same there's one small but important difference which is that the less than signs become less than or equal okay so the difference between the two is that the close interval the as a set it includes these two endpoints right a and b are elements of this set they belong to the set whereas for the open interval they don't right so every number that's in between no matter how close you get to a anything that's big bigger than a no matter how close is in the set anything that's smaller than b no matter how close to b is in the set but a and b themselves are not okay you can also depict this using a using a number line so if we have our real numbers here and let's say a is there and b is there so we want to indicate that where we've got everything in between so we might color that part in actually let's get some better contrast so we color it in like so okay and then we use these kind of hollow circles at those two points to indicate that the a and b are not included okay for a closed interval you kind of start the same mark off a you mark off b but this time to indicate that you're including those points you put a solid dot so you fill it in and you fill in all the points in between okay so those are the two most basic types of intervals open and closed but you can also kind of mash these up right there there are several ones that you might refer to as the half open intervals okay so these are going to be ones that look like say a b so that would mean that you're including the left endpoint but not the right endpoint or you could do it the other way a and then b right so you're including the right end point but not the left okay so these these are the half opening jewels these four together are the four types of bounded intervals okay these are the ones that they don't go on forever in either direction at some point you stop right and once you go beyond that point you're not in the interval anymore but there are also a number of so-called infinite intervals okay and these there's a number of these so we could do like this a to infinity with a closed bracket on the a we can do a to infinity with a round bracket on the a okay as you might expect these are going to be all the real numbers x which are so just like here we want x to be bigger than or equal to a but there's no upper limit right there's no restriction here on how big x can get so we're just going to say that x has to be bigger than or equal to a similarly here we can just say that x is strictly greater than a right so here it's bigger than a or possibly equal here just strictly bigger right not including all right we also want to include the possibility that x might be less than a certain number so we could also have and here we use minus infinity right so we always think of kind of minus infinities at this end of the number line plus infinity's at the other end so we can go from infinity to say b or minus infinity rather up to b and again we can do either closed or open for those you know exactly what's coming here all right so that's going to be all the real numbers x where x is less than or equal to b or strictly less than b okay there's there's one last type of interval that you might see which is that you don't put any limits at all so you just go from minus infinity all the way to infinity and a simpler way to write that down is to simply say well you know what that's all the real numbers so let's just write r okay so in the last video we introduced these nine types of interval that you can have right the open the close the two types of half open all the ones that involve infinity for the most part that's going to do the job for any sorts of sets of real numbers that you need to describe it's probably going to be described using one of these but there are times where you might need to also combine intervals right so you might need to combine more than one type of interval um and an example might be something like this so maybe maybe somebody says all right what i want you to do is i want you to solve an inequality like x squared is bigger than four okay and you think about this for a bit you know okay so what numbers what numbers can i write down whose square is bigger than four we're going to introduce formal techniques for doing this later on once we talk about polynomials and solving polynomial inequalities we'll see that there is there is an approach as sort of a systematic way of doing this but for now let's just think about it um what are some numbers i know whose square is bigger than 4 well maybe we start with thinking about what are the numbers whose square is equal to 4 2 squared is 4 right so the square of 2 is 4. anything that's bigger than 2 if i square it i'm going to get something that's bigger than 4. so certainly if x is bigger than 2 its square will be bigger than 4. is that the only possibility well there's one other solution to the equation right what's the other value of x whose square is equal to 4 the other one is minus 2 okay so let's think about it do we want x to be bigger than minus 2 or smaller than -2 if we want the square to be bigger than 4. well when you square the negative goes away so what we really care about here is the magnitude right and as the magnitude of a negative number increases the actual size of that number as a real number goes down so we would want x to be less than minus 2 right so number is less than minus 2 right minus 3 minus 4 right minus minus 7.5 you know those are all in there right so we could have one of those two possibilities and now you think about how do you write those as intervals well so we would say that we could have the interval from 2 to infinity or we could have the integral from minus infinity up to minus 2. one of the ways that you might write this is you might use this union notation so you might write this as so we might say like this minus infinity to minus 2. most people tend to write the smaller numbers first although this doesn't actually matter you can put it in either order union 2 [Music] to infinity okay so this is just sort of a it's a stylized u and it stands for union okay on the number line what this would look like on a number line if you were going to write things out is you would be saying well what i'm doing is i'm including all the numbers starting at 2 and heading off to infinity as well as all the numbers that are less than -2 so heading off that way right so you have two pieces like that another another way you can combine two intervals right you might be familiar with union as one of two set operations that gets used right another operation that gets used is intersection you'll find intersection a little bit less common in calculus text when you're dealing with unions and there's a simple reason for that intersection gives you all the things that are common to two sets so let's say somebody asks you to simplify the intersection of the interval from one to three with the interval from minus one to 2 okay so if you think about that on the number line okay minus 1 0 1 two three we have two intervals uh one interval is here and let me just kind of offset things so we can see them one of them is there okay the other one is there okay and you want to when you're doing intersection you're keeping all the numbers that belong to both and all the numbers that belong to both are between here and there right it's all the numbers which are bigger than or equal to one but smaller than two so this would be one 2 and you'll notice that when you intersect two intervals you get another interval it might be an interval of a different type but it's still an interval right so you usually don't see intersection notation used very often in calculus textbooks because if you you know most of the sets you're working with are intervals if you intersect two intervals you get another interval and most people are going to choose to just write this rather than the more complicated expression you see on the left all right in this video we're going to briefly talk about absolute value so most people encounter this notion of absolute value at some point in in high school right so the the absolute value the notation is a couple of vertical bars right so we'd write this as absolute value of x where x is some real number okay so what does that mean well there's a number of ways of interpreting absolute value one of the simplest ways is think of this as distance right we can think of this as the distance from zero zero is kind of this this touchstone on the real number line right it's this is it's in the middle it's this reference point it's the distance from zero to x that's one way of describing it okay so let's think of that number line right here's our number line okay there it is we've got to mark zero where is zero somewhere here in the middle okay so you choose any other number and you think about how far away from zero it is so let's say i choose a number out here like i don't know 29 okay that's my number so what's the distance from 0 to 29 right well we measured distance using real numbers right uh the distance from 0 to 29 is simply 29 fair enough right and in fact in general if x is bigger than zero then the absolute value of x is just x right actually we can do one better um what's the distance from zero to zero well zero is a distance of zero from itself right any number is a distance of zero from itself so we can actually say that if x is bigger than or equal to zero the absolute value of x is just x okay now what if we wanted the distance from zero to i don't know minus minus seventeen okay minus 17.5 why not throw a decimal in how far away are they well the distance is is going to be again it's going to be the number right the distance from 0 to any number is that number except we never want distance to be negative right so if we if we throw a negative number in and we want to give the distance just remove the minus sign okay so absolute value of negative 17.5 is positive 17.5 right so what that tells me is that if x is less than zero the absolute value of x is now this will throw some people off right how do i get 17.5 for minus 17.5 while i throw away the minus sign how do i get rid of the minus sign well um one of these basic algebra rules that you learn somewhere along the way is that if you take the negative of a negative it becomes positive right so the way to get a positive number from a negative number is to put a minus sign up front okay some people get thrown off by this because they see that minus sign and every time you see a minus sign you think negative so like wait a sec absolute value is supposed to be positive but over here there's this looks like a negative number but it's not right because x i mean x here is just some variable or it could be any real number and real numbers can be negative so if this happens to be a negative real number and you put a minus sign out front it's going to become positive so even though there's a minus sign there the number might still be positive if the number you started with is negative okay so this is a simple way of thinking about it right it's just distance right and you can generalize this you know if you wanted to talk about the distance from let's say a to b where a and b are just real numbers what's that going to be so if you want to think about how far apart two numbers are you should take their difference right the difference of two numbers tells you how far apart they are the problem is i didn't say here that a was you know bigger or smaller than b right i don't know a might be bigger than b might be smaller than b depending on which of these two numbers is bigger this difference could be positive or negative and the way you get around that is you just slap absolute values on it right as long as you take the absolute value it's going to be a positive number okay so that's the basic idea of absolute value is it just guarantees that everything is positive right you can think in terms of of the graph if you like we'll be doing a lot of graphing as we move forward okay so what these two tell me here is that for positive values of x absolute value of x is just x right so if i'm if i'm graphing if i'm doing the graph y equals absolute value of x when x is bigger than 0 we just graph y equals x when x is smaller than 0 we graph y equals minus x right and you'll you can see from the graph that it never drops below the y-axis or below the x-axis right it's always positive um so which which of course we understood because we defined this thing as a distance it should never be negative distances aren't negative um so this is um these are some basic ideas around absolute value by the way this type of function that is given by one formula and one region and a different formula in another region this is this is what's known as a piecewise function right so what this tells me is the absolute value of x is so-called it's piecewise all right or or you might see this as piecewise defined sometimes to shorten things we just say piecewise or we might be more precise and say piecewise defined right so it's defined in pieces right this piece is defined one way this piece is defined another way okay so of course if you wanted to graph something like that well you just draw the graphs of the appropriate pieces on the appropriate intervals right bigger than zero or less than zero in this case and you've got your picture we'll be seeing plenty of piecewise defined functions later on once we move to talking about limits and continuity so before we move on from absolute value let's say a few words about inequalities okay so the the simplest type of inequality that you encounter in the calculus course are the linear inequalities right so something like minus 3x plus 4 is less than let's say 2. okay so there's there's a typical linear inequality so basic rules for inequalities say well you can always add something to both sides of the inequality that doesn't affect things right so the first thing that you might do here add or subtract right it's nice to get all the numbers on one side so you might decide hey let's let's subtract 4. subtract 4 from both sides right 4 minus 4 they cancel out minus 3x is less than so 2 subtract 4 minus 2. okay now the one mistake you got to watch out for right the thing the mistake that people tend to make um here is is in division now you are also allowed to multiply and divide both sides of inequality by any number except zero right you can never divide by zero but also multiplying both sides by zero destroys the inequality and also it would lead you with nonsense right if you multiply both sides by zero you just get zero is less than zero that's not true right but any other number you can throw in in particular you might decide that you want to divide both sides by minus 3 to get rid of that minus 3 and solve for x but you got to watch out right because if you divide or multiply by a negative number flips that thing around so you got to be careful one way to see that is is you know there's there's you can divide by -3 divide by minus 3. as long as you remember that that's going to flip the sign and that's going to leave you with x bigger then those negatives cancel two-thirds okay so that's one way to do it uh another way to do it is you could you know rather than doing it like that you could say let's move the two to that side move the 3x to that side right because you never have to worry about you know negatives when you're adding and subtracting so if i add 2 to both sides now i've got a 2 on the left if i add 3x to both sides i have 3x on the right and i'm allowed to divide by 3 3 is a positive number and i get 2 3 less than x okay fine those are the same thing right 2 3 less than x is the same statement as x is bigger than 2 3 those are equivalent now you might find yourself having to deal with absolute value inequalities okay you'll especially find yourself having to deal with absolute value inequalities if you're in a calculus course that covers the precise definition of the limit okay there are lots of inequalities involving absolute values because they're part of the definition of the limit if you're going through that formal epsilon delta definition of the limit you're going to see a lot of inequalities with absolute values so there are two basic rules okay and and remember that we're thinking of absolute value in terms of distance right so if i say that absolute value of a is less than b okay where b here is a is a positive real number what does that mean well remember that absolute value is distance from 0 to x right so if i say that so if i say something like the absolute value of x is less than let's say 4. what am i telling you well i'm telling you that that x is the distance from 0 to x is no bigger than 4. and that means that all the numbers 1 2 3 4 units right it's not to scale all right so all the numbers from here to there but not including four those are included those all have a distance from zero that's less than four but that's not all the numbers right because i could also go four units to the left and get to minus four right that would also work right so so this turns out to be the same thing as saying that x is between -4 and 4. and it turns out that this is this is true in general this is the same thing as saying minus b is less than a is less than b okay so a is somewhere between minus b and b and of course this remains true if you replace less than by less than or equal okay the other one is well what if it's the other way around what if we have absolute value of a bigger than b what does that mean well that means you know let's come back to this example right what if i turned it around and said okay i want all the values of x where the absolute value is bigger than four right that means the distance from zero has to be more than four right so i'm not including these numbers in fact i'm including everything else so it's everything outside that interval so that would mean everything that's bigger than 4 or smaller than -4 right so this would be the same as saying that a is bigger than b or a is less than minus b okay and again if you have equals equals there right in terms of intervals that means that you're doing here you'd be going from minus b to b here this is one of these places where you need union right this is going to be from b to infinity but also from minus infinity to minus b okay the context that you're probably going to see a lot is you're going to be measuring the distance between two numbers so you're going to you're going to have a lot of things where you have to rewrite something like absolute value of x minus 3 is less than 2. okay that's going to come up something like this comes up quite a bit so first step apply this rule right think of this x minus 3 as your a think of this as a single unit 2 is your b so we're going to get minus 2 less than x minus 3. less than 2. okay so now you've got one of these compound inequalities that's fine you can add something to each part of a compound inequality and in particular the thing that we want to add we want to get rid of that minus 3. so we add a 3. so minus 2 plus 3 becomes 1 less than x less than 2 plus 3 which is 5. all right so those are the two steps for solving this sort of inequality the other thing that might complicate things is there might be a coefficient out front right but you handle that the same way you handled it in the in the linear case all right so now we come to everyone's favorite middle school stumbling block arithmetic with fractions um so we're going to start with addition which tends to be the trickier part or you know subtract so subtraction is the same thing right subtraction is just addition where the numbers have opposite signs right one of the numbers happens to be negative so if you understand addition you understand a subtraction a lot of people when they're when they're telling you about fraction addition they're going to give you some formula right they're going to say something like well here i'm going to be one of those people that gives it to you uh maybe i won't a over b plus c over d and then they say oh yeah so there's this thing right oh hear why not i'll give it to you they'll tell us a times d plus b times c over b time no no but nobody sits down and memorizes this formula and applies it every time they want to add a couple of fractions right the key to adding fractions is this bit right here right it's the so-called common denominator all right so how does this work so the idea is that when you're adding things you can really only add things that are of the same type think of the denominator and a fraction as as being something like a like a unit right so really when you when you write down a fraction like say say three fifths right well some of this comes back to just how do you interpret fractions how do you visualize fractions um maybe you have this kind of like pie picture right um probably one of the better ways to to think about it is start with uh you know start with the unit interval all right all the real numbers going from zero to one okay and of course if you if you want to do improper fractions or negative fractions you're going to have to modify this slightly but for for fractions between zero and one right the denominator tells you how many pieces you should cut your interval into those are supposed to be five equal pieces not my best effort the numerator tells you how many of those pieces you should keep right so we should keep three of them right so we we can kind of have this sort of picture of our fraction where we we shade three out of five boxes okay so you it always makes sense to add things that are the same right you can add like things so if somebody says i want you to do you know two-fifths plus plus one-fifth you say okay i'm adding two fractions of the same type they're both fifths right so if i had two of these all right and then i added one more i'd have three of them that's fine right where things get more complicated is when somebody says okay i've got two fifths and i want you to add you know let's say three-tenths okay now these are fractions that are are different types right so so this is now you know rather than saying okay i want you to like take two apples and add one more apple um sort of like well it's not quite apples and oranges maybe something like okay someone says i want you to take i want you to take two feet and add on three inches right the answer is not going to be 5 feet it's not going to be 5 inches right it's it's somewhere in between because you need to do a conversion you need to do a unit conversion and so the key is that you know if i if i have something that's divided into five pieces well there's an easy way to get 10 pieces i take each of the five pieces that i have and i divide it in half now i have 10 pieces right and so then my my three-fifths that i had before becomes one two three four five six pieces or in this case two-fifths would become one two three four of the ten pieces right so two-fifths this is the same as four-tenths right and and the way you do the conversion this notion of equivalent fractions is that you can always multiply the numerator and the denominator by the same thing without changing it so we go 2 over 2 right so we have 2 times 2 2 times 5. so 2 times 2 gives me that 4 2 times 5 gives me the 10. we add the 3 tenths and now that we have two fractions of the same type right we can add them okay the most complicated scenario when you're adding numbers right is when you've got two different denominators and neither one is a multiple of the other so you're adding something like one third and you're going to add i don't know let's do one third plus one quarter okay um so if you want to you can think about this visually right you can think about that you've got one piece and you've divided into three pieces you've got another one you've divided into four pieces and you're you're taking one of them and one of them and you're just kind of gonna stick them together right you take that piece and that piece and you're gonna stick them together now you're wondering like how long is that piece right as a as a fraction of the whole how much do i actually have well you do the same equivalent fractions idea but now you're going to have to adjust both denominators until you get one that matches right and that's where this this rule comes in right that one way that you can always get a common denominator is just to multiply the two denominators that you have right 3 times 4 gives me 12. right so i can write both of these as fractions over over 12. i just have to think about what do i need to multiply each one by top and bottom to get to the 12. well 12 is 3 times 4 so here i need 3 times 4. but anything you do to the bottom you should also do to the top right here i need 4 times 3 okay and again do the same thing top and bottom so what i get is now four twelfths plus three twelfths leaving me with seven twelfths okay and then you got it done uh again if you wanted to kind of you know use this sort of picture to visualize what you're doing is each of these three pieces you're dividing into four each of these four pieces you're dividing into three and now each of these bars has been divided into twelve equal pieces right and there are one two three four pieces here one two three pieces here and so you'd have seven pieces there out of the 12 in total all right that's the idea with addition again subtraction if i did if i was doing let's say one third minus a quarter well this would be 4 12 minus 3 12 i get 1 12. right subtraction is not any harder we're going to pause there we're going to come back we're going to do one example with some some variables in it and then we'll talk multiplication all right so one more video on fraction arithmetic this time we're going to look at multiplication right of the two multiplication is actually a little bit easier than addition right because when you're multiplying there's no need to worry about common denominators or anything like this it's simply a matter of multiplying across so again if i were going to give you a formula it might look something like this a over b multiplied by c over d i'm just going to do a times c divided by b times d okay that's the rule for multiplication um i guess i guess maybe one you know some people tend to get these two rules mixed up and maybe one of the reasons that you might tend to get them mixed up is that the denominator ends up the same in both cases which is maybe yet another reason why you just shouldn't try to remember these formulas in the first place right don't don't rely on the formulas maybe i shouldn't even give them to you better to just think about what's going on right so think about doing something like say two-thirds times i don't know three quarters okay something like that and now there's a few different ways that you can you can do this right let's we could blindly apply the rule so if we apply the rule we get 2 times 3 is 6. 3 times 4 is 12. because there's 6 over 12 that's our answer right but that's an unnecessarily complicated fraction because 6 over 12 we know that this can reduce down to one-half all right and then you might be left wondering like oh did i really maybe i did more work than i needed to was there a way to save myself some trouble uh and the answer is yes right you can reduce after like this if you want but you can actually reduce ahead of time right you know that you're going to be combining these into a single fraction okay you know that you can reduce you know that you can cancel factors right if things are factored out top and bottom you can cancel them and so when you see that there's a three here and a three here oh we can say three three over three it's just one over one right two over four two over four is the same as one over two so i can do rather than doing two thirds times three quarters i just do one times a half i get to my answer of one half right so we can we can clean things up like that um this certainly becomes relevant if you're if you're getting to the point where you're multiplying algebraic expressions right you've got x squared over x plus 2 and you need to multiply by x squared plus 4x plus 4 over x squared plus 3x something like that you can multiply it all out if you want right multiply everything on the top by x squared multiply those two binomials on the bottom foil it out then think about simplifying right but once you've multiplied those things out right once you've got this like degree four polynomial up top right degree 3 on the bottom those are going to be hard it's going to be hard to see how to simplify right it's easier now well things are already sort of factored and so rather than multiplying first and then seeing if you can simplify factor first see if you can cancel then move on with the standard node of caution you can only cancel things that have been factored out as common multiples right if you have the same multiple on the top that you do on the bottom you can cancel them if they're terms right factors can be cancelled terms cannot so there's just because there's an x squared here and here i can't cancel them out right that would be like if i had you know if i had if i had i don't know 3 over 4 and i said oh well 3 is 1 plus 2 and 4 is is is 2 plus 2 right i can't cancel the twos and say that's a half right 3 4 is not the same thing as a half right so i can't cancel things that are being added right if there's a plus sign in front of it or behind it you can't cancel right on the other hand if i had something like 6 8 right and i said oh well that's 3 times 2 over over 4 times 2. and i said okay now that 2 right if it's being multiplied same thing top and bottom was multiplying then i can cancel so what i have to do in something like this is first thing i have to do is say okay can i factor and i can so x squared so maybe what we do we can combine them if you like but rather than multiplying through think about factoring this is the square of x plus 2 right on the bottom x plus 2 and then i have x times x plus 3 when i factor that okay and and so then i look to see what is some of the common stuff that i can take out from the top and the bottom both the top and the bottom have an x and they both have an x plus 2 right so i can do x times x plus 2 x times x plus 2 and then write down everything that's left over there's another x and another x plus 2 on the top and on the bottom there's an x plus 3 right so i can i can do that and then realize that yes these these x's these can be canceled those x plus 2's those can be cancelled because i'm multiplying all right and then i've got my simplified answer x x plus 2 x plus 3. okay one more fraction video before we wrap um we do have to talk about division because division it's not quite the same as you know subtraction once you understand addition you understand subtraction they're basically the same thing division adds in one further wrinkles so we should address that in a separate video okay so we're ready to wrap up fraction arithmetic the last topic to tackle is division now you've probably learned the rule of this mantra at some point right that division is the same thing as multiplying by the reciprocal all right so you you flip and multiply some people like to say all right so if you're if you're giving a formula and again i'm questioning whether we really should but here we go a over b divided by c over d you flip the second and you multiply that's the rule so for example if i was going to do something like two-thirds and i wanted to divide by um let's say i don't know um four-ninths okay how do we do that well maybe we should spend some time if we had more time we might try to think a little bit conceptually about how this makes sense right there are lots of ways to understand why this rule works okay if you if you think of division as canceling a multiplication right then this kind of makes sense right c over d and d over c they're opposites of each other in the sense that if you multiply them you get one okay um more basic than that you can go into thinking of division as counting like how many times does one number go into another things like that right now but these kind of basic kind of getting into these core concepts of fractions and fraction addition is i think maybe a little lower than we can go into calculus review there's lots of good material out there on the web if you wanted to do it but i think this is a good one to look at because you'll notice that you kind of when you're doing division the way we tend to write things in like calculus you get these compound fractions right fractions within a fraction right the numerator and the denominator of this fraction are themselves fractions and you have to think about how do you simplify when you write this out what goes where so rule says well what you should do is multiply by this the reciprocal of the one on the bottom okay so instead of dividing by 4 over 9 you multiply by 9 over 4. okay now you can multiply that out or you can you can again notice that hey there's a bit of simplifying i can do here 9 over 3 is just 3 2 over 4 is a half okay and so the result is is 3 over 2. okay that's not so bad you remember if you remember the rule this is probably one of these cases where um you know it's nice if you have a deeper conceptual understanding of what's going on here might help you later on if you're studying algebra or something like that but if you're just trying to get the calculation done remember the rule where this is going to come up with uh say with calculus is you might be dealing with an expression like the following you might be dealing with something like this 1 over x plus a minus 1 over x divided by a okay and you want to simplify this thing and this really tends to trip people up it messes people up because they're like wait that's a fraction that's a fraction wait but it's inside this bigger fraction and i don't know where that a is supposed to go like where the hell does it go so if you remember that division is multiplying by the reciprocal and if you remember that you know now this might be a real number it might not even be a fraction anymore but the rules still work and and so for any for any real number you can still write down the reciprocal of a real number right um if you have some number x and you want to take its reciprocal it's just 1 over that number so we do here is say okay rather than dividing by a we'll multiply by one over a okay now it becomes a little bit more clear how to proceed if we wanted to simplify okay and this is the sort of thing you're going to have to do this is going to come up when you're using limits to calculate derivatives is actually a fairly standard example that i'm doing for you so the next thing we'd do well we'd have to combine these right again common denominator in this case that common denominator is going to have to be the product so i'm going to have to multiply this one top and bottom by x you have to multiply this one top and bottom by x plus a and coming all the way back to one of our very early videos we talked about order of operations and you probably thought i was being silly because hey you you learned order of operations like way back in elementary school right you don't need to cover that here's another place where people are going to screw up because a lot of people are going to apply this minus sign to the x but not to the a because they don't realize that there's actually again these implied parentheses there we often don't bother to write those parentheses but they're there and it becomes relevant when we go to the next step because in the next step and now we can kind of the whole denominator is going to be the a times the x times the x plus a so we have a times x times x plus a and we have x subtract x plus a right so that minus sign hits both of them right minus x minus a right it's distributed to both of those terms and if you're not careful you might miss the minus sign on the a x minus x cancels you just get negative a over a times x times x plus a and the last thing you might choose to do is realize that this is just minus 1 times a and then say hey i've got an a on the top i've got an a on the bottom so why don't i cancel and get down to a final answer of minus 1 over x times x plus a okay you'll definitely be seeing like calculations like that once you move on once you move on to limits derivatives you're going to be seeing these sorts of things as long as you're careful right and as long as you remember that when you're dividing that really means multiplying by the reciprocal you'll get things in the right place and you'll be okay all right a lot of people you know the two most common pitfalls here might be that the a ends up in the numerator when it should have been in the denominator and you might forget something simple like order of operations distributive property forgetting that there are really brackets there okay all right that's it for fractions um we're going to move on all right so in the next few videos we're going to talk about rules for manipulating exponents i've thrown up a reference here to uh to a blog i like to read sometimes uh math with bad drawings it's not a bad little blog but he has a really great post from a year or two ago titled the exponential bait and switch if you search google on this title you'll find the blog it's good reading and it's it's interesting because it points out something that's i think kind of fundamental to to a lot of mathematics and development of mathematics which is you know when you're starting with exponents how did you first learn exponents right think way back um actually think all the way back to like multiplication right you might have first learned that oh multiplication is a repeated addition and then you learn oh it's not quite so simple because what if the numbers you're multiplying you know are not whole numbers what if what if one of them is negative one of one's a fraction what if you're multiplying by the square root of two how do you make sense of that right and the same thing happens with exponents right somebody says oh exponents are our repeated multiplication right so so they say well you know a a cubed just means you know a times a times a right and a to the 4 just means a times a times a times a and so it stands to reason that all right you want to do a cubed times a to the 4 well that's a times a times a times a times a times a times a and now basic algebraic properties of real numbers take hold right we know that multiplication is is associative these brackets are somehow unnecessary um right and community if i can you know a a cubed times a to the 4 is the same as a 4 a to the 4 times a cubed right i can i mean i'm just multiplying by a the order in which i multiply by a doesn't matter the only thing that matters is how many times did i multiply by a and how many times did i multiply by a well one two three four five six seven times right a to the seven right and so then you generalize this and you say okay well in general right if i wanted to do you know a to the m times a to the n that should be a to the m plus n similar thinking leads to a rule for division right if i if i did a cubed divide by a 4 right i had 3 in the numerator 4 in the denominator i would start canceling off common factors until i had just one left in the bottom and and so you kind of you know it's however many you had on top minus however many you had on the bottom and so well that gets you to this rule right you subtract the exponents um but then of course that that leads you to uh to a possibility right if i was doing a a cube divided by a to the four right i've got more a's on the bottom than i do on the top this might be negative so then you say oh what what if i have a negative exponent right what if i have like a to the minus n uh well there's a simple rule for that too right negative exponents are just reciprocals right and so you sort of you start with this idea that exponentiation is repeated multiplication um and it leads you to these rules and then eventually you say actually the rules are all that matters right you should probably remember where it came from but ultimately it's the rules that matter right and you say well exponents are just the things that follow these rules and and then that lets you to lets you extend things to scenarios where maybe maybe this exponent is not an integer maybe it's a real number right maybe it's a fraction right maybe we're working with our favorite base right e we might be doing that so we're going to we're going to be looking at situations where maybe the the base and the exponent are are real numbers they might be irrational numbers we still need to make sense of this and you make sense of it through the properties right through the rules um there's one that i've missed right which is well there's a couple but another one is that if i had a to the m and i was going to raise it to a further power and in this case you multiply the exponents and i guess that comes down to the whole you know multiplication is repeated addition right because this means i multiply a to the m by itself n times i apply this rule which is that i should add m to itself n times which really means i should multiply m by n right so you can you can kind of put all those together there's another one that comes up maybe less often for calculus but i suppose it still comes up you might be dealing with cases where you have either a product or a quotient that you want to raise to a power in this case you can distribute the exponent to both terms right a to the n over b to the n okay those rules work those rules are highly dependent on the fact that the order of multiplication doesn't matter for real numbers right because on this side you're doing a times b times a times b times a times b times a times b times a times b on that side you're doing a times a times a times a times a and then b times b times b times b right you you had to move all the a's to the beginning and all the b's to the end right you had to rearrange um if you were doing another course let's say like linear algebra where you're dealing with say matrices where the order of multiplication does matter you've got to be careful about these things but we're doing calculus we're working with real numbers so it's okay we have these rules they work all right so we can play with those probably the last one to point out is is fractional exponents right what um the other one is is to note that if i had a to say the 1 over n right that means the same thing as the nth root of a right and and that kind of you know in a way that makes sense um kind of thinking about this rule thinking about that rule that the nth root of a should be the thing that if you multiply you know it by itself n times or even the nth root of a times nth root of a n times i should end up with a right but if i do a to the 1 over n times 1 over n times 1 over n right n of those right i should get if i add 1 over n n times i get 1 i get back a it makes sense right and the last one is if i have a rational exponent so not just something where the numerator is 1 but a general rational exponent you can write this as either you do a to the m and then you take the nth root or you could take the nth root and then raise it to the power m and you'll get the same result either way those are the basic rules for exponents we'll look at some examples in the next video all right so in the last video we went over these uh rules for working with exponents now we should see how these work in practice okay so in the first example we're just going to point out some some basic principles here this is coming all the way back again to the stuff that you you thought you were too old for right order of operations um you know brackets exponents things like that uh because again it's you know we think we know it but we mess it up all the time so so one is to think about you know if i had like two plus 3 squared right versus say 2 plus 3 squared these are not the same thing right because order of operations says that we should always do the exponent first right this one is is 2 plus 9 right which is 11 right because you apply the square first and then you add two whereas here the parentheses say well oh i should really first do two plus three i do the thing inside the bracket first i do the five then i square it and i get 25 right maybe the other one we should throw in here right because this is one that that again is is a common mistake maybe not when there's numbers in but definitely this is a once there's an x in there it's going to happen 2 squared plus 3 squared right so there's there's always this tendency there's this you know this wishful thinking we want to just apply the exponent to both things right but we know that this this is not right what does that give me 2 squared is 4. 3 squared is 9. 4 plus 9 is 13. right all very different answers right of course the reason that these ones don't agree is that when you're doing 2 plus 3 squared right again think in terms of if you like repeated multiplication this really means 2 plus 3 times 2 plus 3. and if you wanted to you could distribute that out right nobody would ever bother when it's just numbers but say say this was x plus 3 right if it's x plus 3 well then we know what to do right x plus 3 we want to square it x plus 3 times x plus 3 and and we multiply that out now most of you probably kind of internalized this you've remembered your your foil rule you might have even memorized the formula for for a square right you can just write down the answer without even thinking about it certainly saves time but one thing to remember is already you're doing is you're doing the first term here right x times the bracket x plus three then you're doing the three times that bracket x plus three and then you're expanding again x times x x squared x times three three x three times x three x three times three nine right lots of people say oh yeah this is foil rule right first outside inside last they get straight to that answer that's fine too uh the only downside with with kind of you know relying on something like foil is is you know what if what if there was another term here what if there was another x plus three what if this was x squared plus two x plus three right what if one of these had three terms instead of two foil's nice but it only deals with one situation among many it's a common situation that's why we have an acronym but you know there are lots of other situations that can come up so it's nice to remember you know you can do this even if you forget foil last step of course combine the middle terms a lot of you probably can write down that answer in one step that's fine if you can but if you've forgotten you're out of practice you can always get there by relying on fundamentals right okay so it's important to remember things like this it's not just x squared plus nine there's that term in the middle right it's easy to forget it um the other one where you might forget it is you know what if i'm doing something like uh square root of 1 plus 3. this is another one of these cases where you have this this idea that there there are implied parentheses right the 1 plus 3 is inside the square root right it's inside this if you like a square root function we haven't quite talked about functions yet but you ought to do the addition before you do the square root right this is the square root of four it's two right it's not square root of 1 plus the square root of 3 right 1 plus root 3 it's definitely not the same thing as 2. one is an integer the other one's not even rational okay so be careful about that everyone says oh yeah yeah of course i know i would never do that yeah you say you would never do that now and then i give you something like the square root of x squared plus 4. and i know somebody is going to tell you that's x plus 2. it's not right this is just something that does not simplify okay sometimes we just want to simplify things we want we want our answers to be as simple as possible and sometimes they can't be as simple as we'd like it to be and that's okay that's something you just kind of have to live with um okay um some basic examples with uh with exponents i think we're gonna we're gonna pause here we're gonna come back we're gonna do some uh some slightly more complicated examples before we move on all right so as promised here are a few examples working with laws of exponents i've left the laws up here in case we need to refer to them we'll start simple we'll work our way up you might think that the one at the end with numbers should be the simplest everything else has variables but that's an example that messes people up all the time people really struggle with that one so let's have a look first one x cubed times x to the sixth well that's a straightforward application of this first rule right we're multiplying two powers with the same base so we simply add the exponents three plus six is 9 that's x to the 9. now i've thrown the other one in because people tend to get mixed up on these right when should i when do i use that rule right the rule applies for multiplication there is no corresponding rule for addition right so there's nothing you can do really to combine those two terms so you leave them as is right the only thing you could possibly do to simplify this is is to kind of use this rule backwards you could say well x to the sixth that's that's x cubed times x cubed which might come in handy because you might want to factor this right so you might want to say oh that's x cubed times so x cubed is x cubed times 1 right and then x to the 6 is x cubed times times x cubed right so you might be able to factor but you can't combine right you can't there's nothing you can do to get rid of that plus sign it's going to be there whatever form you write it in that plus sign is going to stick around okay let's look at these ones all right so this one here we need two different rules it's combination of this rule here for raising a power to a power as long as as well as what to do when you have two different numbers in the base all right so let's apply this rule first right so we're going to distribute the 4 to the two different bases so this is going to be x cubed to the fourth power times y squared to the fourth power and now we multiply the exponents so 3 times 4 is 12. 2 times 4 is 8. and again i don't know what x and y are so there's nothing i can do to combine those i have to leave it as is now the next one i've got a square root how do you deal with that well the key to dealing with this one and simplifying if you can is remembering that square roots can be written as fractional powers okay so i could write this as x cubed y squared okay to the power one half all right now i can distribute the power the same as before x cubed to the one-half y squared to the one-half okay there's actually something subtle here when you're simplifying okay here we want to we want to apply the rule right we multiply this rule so we say okay x cubed to the one-half well that's x to the three halves and that's fine you're you're more or less using this rule here when you do that right you're saying the square root of x cubed i can write as x to the 3 half right it doesn't matter whether i do the square root first or the cube first i'm going to get that answer okay y squared to the one-half now here there's something that you have to watch out for the temptation is to just write y right the square root of a square doesn't always give you back the thing that you started with right because what if the thing you started with was a negative number right let's say y is minus 3 what happens when you square minus 3. i square minus 3 i get plus 9. what's the square root of 9 it's 3 not minus 3 it's 3. so i don't actually get back the thing i started with turns out what you get is the absolute value so there's there's a general rule here which says that the square root of a square is the absolute value okay so you have to be a little bit careful with that right um i'm being slightly lazy when i write down this rule here really i should be careful because m and n if they're let's say fractional powers there's not always going to be necessarily this agreement on this on both sides really what i should say here probably is that either a has to be positive or m and n have to be integer exponents um so you gotta be a little bit careful about some of these things all right how about this one well there's two ways to think about it one is to just sort of apply the rule here directly right x squared so it's 2 minus minus 3 x to the fifth okay it's fine the other thing you might do if you if you wanted to is to say well that's the same thing right if the double negative is throwing you off it's remember that what you've got here is x squared times so if you've got a negative exponent you can bring it upstairs right um so this rule here could be rewritten and you know if i put minus minus here i'd have the minus there right so i could move that minus to the other side 1 over x to the minus 3 is the same thing as x to the 3. okay so a negative exponent on the bottom becomes a positive exponent on top all right this one here what do we do well there's two choices right and again this is one of these kind of it's an order of operations question and it's one where in this case it doesn't matter you'll get the same result either way we can either first apply the outside power to everything so cube all four terms then see if we can simplify or we might want to simplify inside first and then apply the power generally speaking it's easier to simplify first right you want to simplify and then and then apply powers i expanding and then trying to simplify is usually a little bit trickier so x over root x that's x to the one remember that this is x to the one half one minus a half gives me a half right x over root x is just root x and same as up here y squared over y to the minus 1. i bring that up 2 plus 1 gives me y cubed i still have to cube so x to the three halves three times three gives me y to the nine okay how about this last one so this one is tricky we've got the fractional power negative exponent numbers inside how do we deal well the first thing is we might notice that 8 and 27 are themselves cubes we can write them as powers one thing we might want to do maybe before we do anything let's get rid of that negative so the negative exponent means reciprocal right so i can get rid of the negative exponent by flipping my fraction that gets rid of the minus sign but still the two thirds be careful you don't flip that fraction you're flipping that fraction okay then i might realize that 27 is three cubed eight is two cubed okay and i can apply that power to both of them two thirds two thirds now we use this rule right three times two-thirds three times two over three just leaves me with two so three squared over same logic two squared so my result is nine over four all right so in this video we're going to talk about lines right so this is a very elementary thing that you've been seeing for years um and it seems really really basic but one of the reasons that lines seem really really basic is that in your school career this is something that was developed over several grades you didn't learn about lines all at once believe it or not it took some time to develop the idea of just simply a straight line so how do we think about lines how is the line defined well the idea of a line goes all the way back to basic kind of euclidean geometry right going all the way back to the ancient greeks so one of the things we can do is is we can talk about a line segment right so between any two points we can connect them by a line segment right something like that it's not yet a line a line is what you get if you take a line segment and then this is one of the basic kind of axioms of euclidean geometry that if you have such a segment you can extend it off to infinity in either direction and then you've got yourself a line okay so there's a line so what is it that makes lines special what is it that sets a line apart from other curves that we could draw in the plane well one of the defining characteristics of a line is slope right so you've probably thought of slope as this idea of rise over run right so as a formula we might give it as change in y over change in x right so for these two points i've indicated on my line we might write this slope as what's the change in y well we have we can kind of draw a little right angled triangle here okay the height right is this change so here this point here that i've drawn the i've changed the x coordinate but not the y coordinate right so the x coordinate has changed to c the y coordinate is still b so the change in y right is this change between b and d d minus b okay that's the change in y the change in x you can see along here right the difference in x is c minus a right there's the slope for our line okay that's fine but i can you know i can i could do this for any two points on any curve right i could draw a parabola circle i can choose two points i can calculate delta y over delta x right for any curve that i want what's special about a line well the the thing that's special about a line what makes lines different from other curves is that this number is constant i will get the same slope for any two points on the line that i choose so if i were to pick some other point let's say here and call it x y if i calculate delta y over delta x using these two points or using these two points i'll get the same number that i got when i did those two points all right so that means that my slope could also be written as y minus b right over over x minus a and it could also be written as y minus d over x minus c right those should give me the same points and and with a bit of manipulating here you can actually get the equation of the line right if i multiply both sides by x minus a what i get is i get y minus b is equal to m times x minus a or if you like y minus d equals m times x minus c right and you might be concerned that those those appear to be different right i've got i've got an a and a b here i've got a c and a d there so i've got two different equations for my line right because i've you know essentially i've written down these equations using one point or another point and the point if you like is that it doesn't matter you'll get the same answer either way because in this case if i kind of if i were to simplify let's say i solve for y so y is going to be so there's this m times x let's put that in and then i've got minus m a plus b okay over here y is m times x minus m c right plus d okay so we want to believe that these are the same numbers but that comes up to you know here was i this equation to begin with right so m times c minus a is equal to d minus b so m c minus m a is equal to d minus b okay or if you like moving that over and moving that over b minus m a is equal to d minus m c b minus m a equals d minus m c right so it's the same line right it doesn't matter which point you use this form here this equation by the way this is usually called the point slope form of a lie a lot of you are probably more used to seeing it in this form here right so m x plus and then let's just put that all b minus m a this is sometimes called the slope intercept form right because this number here this b minus m a right that's the y intercept for your line right um i know um you're probably a little bit concerned because i'm using this b and this b is not the intercept right you use m x plus b b is the intercept right um b is not set in stone as as being anything right we we can use letters interchangeably um so most of you are probably used to writing it in this slope intercept form and a lot of you are probably going to insist on writing it in the intercept form but it turns out for most purposes in calculus this is the more useful form okay and it's the easier one to get to because what's going to happen most of the time most of the lines that you're going to encounter in calculus are going to be tangent lines they're going to be tangents to curves the information that you get for constructing those lines well there's going to be a slope that slope is going to come from the derivative of some function the derivative is going to tell us about slope the other thing you're going to have is going to be a point because you're constructing that tangent at some point on a curve right so the point and the slope are going to be information that you have you can immediately just plug it into this version of the line and you have your answer right there's no need to go to the trouble of figuring out the y-intercept so that you can do slope-intercept just go with point-slope the other reason that this is more useful on in calculus you'll find that this is exactly what you need if you want to talk about linear approximations right one of the reasons that we construct tangent lines one of the reasons we talk about derivatives is we use them to do approximations this is going to be the more useful form to talk about that all right so in the next couple of videos we're going to look at some basic algebra we're going to look at expanding and factoring when you have variables involved so basic example the one that a lot of people learn is this uh foil rule right when you've got something like 2x minus 1 you want to multiply by x plus 4. so the foil stands for first outside inside last right so you you multiply the first term so you do the 2x times x 2x squared the outside terms the 2x and the 4 4 times 2 gives you 8x minus 1 times x those are the inside ones so minus x and then finally minus 1 times 4 the last terms okay you can simplify in the middle and you're done now that's fine if your problem involves multiplying a couple of binomials then foil works just great but this is not going to be the only situation that you run into right there's going to be other other types of multiplication that you have to do so what's probably better to do is to realize that what you're doing when you're doing foil right is you're relying on this distributive property the fact that you can take this x plus 4 multiply it through these brackets you can do the x plus 4 multiplied by 2x and then do the x plus 4 multiplied by minus 1. so when you take this and you multiply by 2x right that's going to give you this term and that term the first two terms when you take this and you multiply by minus 1 it's giving you the other two terms right so these ones here all right come from multiplying by 2x and that produces this term and that term when you multiply by the minus 1 it's going to produce this term and that term and if you remember this in terms of this basic distributive property then you can extend from foil to other situations so you might be dealing with a problem where you've got x minus 2 and instead of having to multiply by another binomial maybe you've got something like 3x squared minus 2x plus 4 right suddenly foil doesn't work because this is not a binomial so you can't rely on foil but you can still rely on the distributive property you can take this bracket here multiply first by the x then by the minus two so we can do x times three x squared minus two x plus four minus two times three x squared minus two x plus four and then distributive property again push the x through the brackets push the minus two through the brackets don't lose the minus sign so we're going to get 3 x cubed minus 2 x squared plus 4 x and then minus 2 times 3 minus 6. x squared all right minus 2 times minus 2 double negative gives you positive plus 4 x minus 2 times 4 minus 8. all right and if you want you can group terms right so there's only one degree three term x cubed there are two degree two terms we can group those together minus eight x squared two four x's so plus eight x minus eight and you're done okay you can also expand if you have three or more factors that you need to multiply out so maybe you have to do something like x minus two two x plus one 3x minus 4. you want to multiply that out now this is one of these situations where again you're relying on these basic algebraic properties of the real numbers that we don't necessarily always explicitly state but we use frequently one of the multiplication properties for the real numbers is this associative property for multiplication which says if you need to multiply three or more things it doesn't matter how you group you could choose to group these two together so do the product of the first two and then multiply by the third or you could decide that you want to do these ones here multiply the last two and then multiply by the first you can group it either way you'll get the same answer so why don't we decide that we want to group the first ones together right so then we're going to do the multiplication inside the larger brackets first then we're going to deal with that last term so then what we get is so x times 2x 2x squared x times 1. you'll notice i'm guilty of actually using foil here minus 2 times 2x minus 4x minus 2 times 1 minus 2 times 3x minus 4. probably you want to simplify this before you multiply it all out so 2x squared minus 3x minus 2 times 3x minus 4. and again you multiply it all out and we're now in this sort of scenario here right there are three terms so again we can't fall back on foil um but with a bit of practice you can kind of you know you can do this step in your head just like you kind of do with foil right so you can say okay i'm going to take each term in the first bracket first i'm going to multiply by the 3x so 2x squared times 3x i get 6 x cubed minus 3x times 3x minus 9x squared minus 2 times 3x minus 6x okay now i'm going to take those three terms i'm going to multiply them all by minus 4. so 2x squared times a minus 4 minus 8x squared minus 3x times minus 4 minus minus gives me plus 12 x and then finally minus 2 times minus 4 plus 8. so now we group right 6 x cubed minus 9 minus 8 minus x squared minus six plus twelve plus six x and then finally plus eight and you're all done so before we move on to factoring we're going to spend a few more minutes on expanding there's our last example from the previous video we're going to look at some basic formulas and in particular what we're going to look at is the so-called binomial formula all right so there are a few instances of the binomial formula that you're probably familiar with okay in general the binomial formula has to do with expanding all right so in the last video we mentioned this binomial formula right so this this result here is called the binomial theorem or the binomial formula whichever you like and it tells you how to multiply out powers of binomials so things let's say of the form x plus a to the n these show up fairly frequently in calculus so it's nice to know how to do this these these coefficients these numbers that show up in front of each term these are called binomial coefficients you sometimes read these as n choose k and they come up in a number of contexts one of the contexts where these numbers come up and the reason that we read this is and choose k is that these numbers tell us the number of different ways that it is possible to choose k items out of a set of n items right where you don't care about the order in which you choose them um so these sometimes go by the name of you might have heard of permutations and combinations this is the combination side of thing now a lot of people when they're when they're some practice they probably don't necessarily use the definition here to work out the coefficients when they're expanding a lot of people remember this result called pascal's triangle and pascal's triangle gives you a way of organizing these binomial coefficients so pascal's triangle starts with a one at the top and actually it's going to have ones down the sides so one so the next row is a couple of ones the way you proceed for each successive row is you add a one on the outside and in between you look at the two terms that are immediately above and you add them together one plus one is two all right to get to the next row one on the outside one plus two is three two plus one is three one on the outside all right next row one on the outside one plus three is four three plus three is six three plus one is four and a one on the outside all right do one more row so 1 on the outside 1 plus 4 is 5 4 plus 6 is 10 6 plus 4 is 10 4 plus 1 is 5. one on the outside all right and this goes on you can add as many rows as you want right this goes on forever okay now one of the things that you'll notice is that these numbers that you get they are exactly the binomial coefficients right so going across this top level is is kind of k equals zero if you like or n equals zero right so this is zero one right because if you just do x plus a to the 1 you just get x plus a 1 1 right coefficients are 1. in the last video we saw 100x plus a squared right x squared plus 2ax plus a squared all right so that's what you get for squaring here we recognize the coefficients we did this one right for a cube one three three one and the next row is fourth power fifth power and so on okay so for example let's say somebody asks you to do oh let's say they want you to do x plus two and they want it to the sixth power okay ah well we didn't go as far as six right so we'd have to do one more row one one plus five six fifteen ten plus ten twenty fifteen you notice there's symmetry so once i've got up to here i can just repeat going back the other way right i have those coefficients so once i once i have those coefficients i can just write this out i can say okay it's going to be x to the 6 plus 6 times 2 times x to the fifth plus 15 times 2 squared times x to the 4. plus 20 times 2 cubed times x cubed right plus 15 times 2 to the 4 times x squared plus 6 times 2 to the fifth times x and finally the last term is going to be 2 to the 6th okay and and i suppose if you were so inclined you could try to clean up these coefficients i don't know if we if we want to be that energetic but let's give it a shot x to the sixth 12 x to the fifth plus 60 x to the 4 plus 168 x cubed um uh here's the here's the tough one right 16 times 15. well see we just double four times so double 130 60 120 240. check my math on these x squared 6 times 32 that's going to be 192 i think i can check on that times x finally 2 to the 6 64. okay and you're done all right so i mean there's a bit of work involved you gotta you gotta remember the formula maybe remember the triangle a bit of arithmetic here simplifying those coefficients um but compare that to if you had to do x plus two times x plus two times x plus two times x plus 2 times x plus 2 times x plus you know 6 6 factors of x plus 2 and you had to expand the whole thing out you'd be at it for a very long time right there's a reason we like the binomial theorem okay so we're working through some material on expanding and factoring this basic algebra um right that you deal with when you're multiplying out products of of binomials things like this we looked at binomial theorem all of this falls under this general umbrella of of polynomials right so a polynomial expression is something that basically consists of well there are two ingredients right powers of some variable and coefficients right plus addition you're going to add you know you're going to have more than one term right so a polynomial with a single term is called a monomial with two terms binomial we've encountered those right and then trinomial and so on right you can have as many terms as you want in a polynomial and when we are talking polynomials we need to specify that the powers were we're looking at here are integer powers positive integer powers right we're not looking at negative powers we're not looking at fractional powers okay so things that are not polynomial include anything with say like a a 1 over x 1 over x squared things like that okay these are examples of what are called rational functions we'll talk about them later root functions so like square root of of t right cube root of s these are not polynomial functions also any of your any of your more complicated functions like your trig functions your exponential functions your logarithms certainly not polynomial okay so in a polynomial expression we're just looking at integer positive integer powers of some variable with coefficients and that means you're looking at things that look like this so there might be some x to the n and there might be some number out front and then so the way we tend to write this is we put a subscript okay now some people will get thrown off by that subscript because you know we're used to writing superscripts for powers they're like what is that what is that thing down there what does that mean is that some kind of power or is it like a secret power what is it going what is it doing um it's not a power it's just an index it's just a way of keeping track of the coefficients saying this is the coefficient corresponding to the power n okay the issue is i don't know how big this power is right i could use like a b c d e for my coefficients but maybe i want to talk about a polynomial of degree 500 i'm going to run our letters we never run out of numbers so we can index the coefficients right so there'll be a n maybe a n minus one x to the n minus one so on down to an a2 x squared a one times x and finally a sub zero okay so this is a polynomial expression now later on we might want to talk about this as say a polynomial function so we might we might give it a name we might call this say you know p of x think of it as a function but we're not yet at the point where we necessarily need to think in terms of functions we just want to think of these as expressions with a variable because we want to talk about how to manipulate them do things like factoring and we want to introduce some terminology so this this a sub n this is sometimes called the leading coefficient okay a sub zero this is called the constant term right it's the only one that doesn't have the variable in it this number here n assuming that this leading coefficient is non-zero this number n is called the degree of your polynomial okay so those are some basic terms that you'll see associated with polynomials so let's do a few examples so we could write down something like five x cubed minus six x plus eight this is an example of a degree 3 polynomial right because the highest power of x that i see is 3. right there is no x squared term that's fine you don't need to have every single power of x showing up in a polynomial the terms that don't appear are just ones where the coefficient happens to be zero which can happen right but notice that everything is just a power of x or a constant there's no negative powers there's no fractional powers nothing like that right let's do another one minus four thirds x to the seven plus two root two x to the sixth minus i don't know pi squared x to the four minus 2x plus 1. okay that's the polynomial it's a polynomial whose degree is 7 constant term is 1. now you'll notice that well there are some kind of you know questionable looking things here but these are coefficients right root two is a real number right it's okay to have square root of a number right pi it's irrational that's okay you can have irrational coefficients you just don't want to have square root of x square root of 2 is okay square root of x is not okay so these are examples of polynomials right again something like and there are sometimes there are ones that you you want to treat as polynomials but they're not quite um so let's say we do something like x squared minus 1 over x plus 1. now it happens that the top is a so-called difference of squares right i can i can factor and then you might say hey look we've got the same factor top and bottom x plus 1. let's cancel that thing right and so you cancel it out and you say ah that gets me to x minus 1. that's a polynomial right well not exactly because a polynomial is defined for every possible real number value that you want to substitute for x okay if we're thinking as a function with the domain the domain is all real numbers here there's a restriction these are only equal if x is not equal to -1 okay because this over here technically is not defined at minus 1. if i plug in x equals minus 1 i get 0 divided by 0 which doesn't make any sense okay so looks like a polynomial it's not quite right it's actually a rational function okay so there's some basics on polynomials we're going to pause here we're going to come back we're going to give you two fundamental results about polynomials and then we're going to move on to some other topics okay so in the last video we introduced some basic terminology about polynomials in this video we're going to give you two fundamental theorems basic facts about polynomial functions so if i have a function p of x that is in this polynomial form there are two things that i can tell you about such a function first is the factor theorem the factor theorem says the following it says that you take any number so any real number say a and you plug it in to your polynomial if you get zero right so this means you're doing you know a n times a to the n a n minus 1 a you plug in you're plugging in this number if p of a equals 0 then x minus a is a factor of p of x okay so in other words that means that you can write p of x as x minus a times some other polynomial q of x so q would be a polynomial whose degree is is one less than the polynomial you started with in fact this statement is is what a mathematician might call an if and only if statement right if you have this factor then certainly plugging in x equal to a gives you 0 because a minus a gives you 0. right so if you have the factor p of a will be zero but the more useful direction is this one that if you plug in the number and you get zero well then you know you have a factor right so you can get started on factoring this is if you're trying to factor a polynomial this is a key result but it still leaves you with one question how do you know which numbers to plug in right there's lots of possibilities there also there are some other kind of more advanced results that tell you something about where to look for possible numbers you could plug in these numbers they give you 0 by the way they have a name right so this a if p of a gives you 0 you would say that a is a is a root of your polynomial okay so rational roots means you're looking for roots that are rational numbers numbers that can be expressed as fractions right integer over integer so there are some other results that kind of can narrow down your search somewhat but the rational roots theorem tells you that if you're looking for you know nice roots so rational or even better integer roots for your polynomial there are only certain possibilities right so the rational roots says that if a is a root of p of x with a equal to let's say um oh maybe i shouldn't use p m say oh actually m is not great either um i was running out of letters um u over v so u and v here are integers what can i say about these integers u and v well it turns out that at least you know i guess we should probably say that we're in lowest terms here because we could multiply and make u and v really big uh if u and v are in lowest terms um let's specify that then you can say something you can say that this a0 the constant term is divisible by u and a n is divisible by v that's one way to say it okay maybe there's nicer ways of saying it but this is one way of saying okay so so the way this works in practice is you're looking for rational roots so what you do is you you look for all the numbers that divide evenly into the constant term you look for all the numbers that divide evenly into the leading term leading coefficient and you use those to form fractions okay and only fractions of that form are possible roots for your polynomial you don't have to consider any other possibilities so let's let's try a quick example and if i just kind of write down a polynomial at random there's always there's always a chance that it doesn't have any rational roots at all okay so let's uh let's go with this example so let's say 3x plus oh i don't know um four okay so possible rational roots what are they um well we know that four 4 is divisible by plus or minus 1 plus or minus 2 plus or minus 4 right 2 divisible by plus or minus 1 plus or minus 2. so that means that the possible roots the ones that you would consider are going to be well i could take plus or minus 1 divided by plus or minus 2. so i could have plus or minus one half i could have plus or minus one over plus or minus one right so plus one minus one those are possibilities i could do plus or minus two over plus so that would be plus or minus 2. if i do 2 over 2 that gives me 1 already got it 4 over 2 gives me 2. already got it so the other options would be plus or minus 4. four so that's a total of eight numbers to consider right which is still a fair amount of work but it's better it's better than you know just randomly guessing right at least at least you've narrowed it down to eight possibilities so how do you figure out if any of those actually work well now you come back to the factor theorem you take each of those numbers you plug them into the polynomial so for example if i wanted to try one i'd come in and say okay so two times one right so i could do p of 1. p of 1 would be 2 minus 1 plus 3 plus 4 and i'd say okay so that works out to eight definitely not zero okay so one's not going to work then i might try p of minus one see where that gets me right and then and then i might try p of two p of minus two p of one half p of minus one half um see if any of them work if none of them work then there aren't any rational roots and that means that well if i was asking you to factor this polynomial i have not given you a fair problem right it doesn't mean there isn't a root it just means that that root is irrational and that means you're not going to be able to find it by elementary methods using factoring anything like that you're not going to be able to find it right it's going to be some ugly irrational thing involving cube roots and square roots and probably the way you would find that is using some numerical techniques that you learn later on in calculus something like newton's method for kind of approximating the value of the root right it would not be fair to ask you to find the exact value for an irrational root yes there is a formula for finding roots of cubes just like the quadratic formula but nobody remembers it nobody uses it you'd try these possibilities if none of them worked you'd move on to something else alright so the next few videos we're going to look at techniques for factoring okay so the ability to factor is fairly important in calculus they're going to be a number of situations where you need to figure out where a function or its derivative or its second derivative is equal to zero right those points where derivative is equal to zero are very important they have a name they're called critical points they come up quite frequently in a lot of optimization and applied problems involving calculus so you need to be able to figure out where functions are equal to zero if they're polynomial or rational functions that's going to involve a certain amount of factoring so basic step is you're looking at quadratics okay so you're looking at something that looks like ax squared plus bx plus c and you want to factor this thing right and and we know from the factory theorem if we find those factors that's the same thing as finding zeros right um so in some sense we're trying to figure out where this where this thing is equal to zero um of course the quadratic formula gives us one answer right so the quadratic formula says oh yeah we know exactly when this thing's going to be equal to 0 it's going to happen when x is equal to plus or minus sorry the um quadratic formula sometimes you forget it minus b plus or minus square root b squared minus four a c over 2a right so some people just say oh yeah remember the quadratic formula it'll give you the answer it's not a nice formula right it's not great but it is a failsafe it's going to work in situations where you can't figure out the factors probably because well there's a couple of possibilities quadratic might be irreducible or it might not have any roots it might not have factors it might be that that's as simplified as you can make it it might be that there are factors but they're they're irrational and you're probably not going to be able to find those just by staring at it um in those cases the quadratic formula can bail you out right so for example if i were to just kind of write down some quadratic equation without thinking too hard about it say okay i want to i want to solve that um probably don't come up with factors right away yeah maybe there aren't any nice factors so if we needed to we can fall back on the quadratic formula we can say oh yeah so the roots are going to be if they exist minus b so minus b is minus five plus or minus square root minus five squared minus four times two times four over two times 2. all right and actually this gives you the answer right away because the first thing you do before you bother with any other simplifying is you look under that square root and you say okay what do i have under that square root under that square root 5 squared is 25 subtract 4 times 4 is 16 times 2 is 32. 25 minus 32 is less than zero okay so i've got a negative number under the square root that tells me there are no real number of solutions right you can't take the square root of a negative number so quadratic formula tells you there are no solutions to this equation so there won't be any factors right factor of theorem tells you that if there's no solutions there's no factors okay so that's one case where you might need the quadratic formula another one might be well let's let's say that this this plus 4 was actually a minus 4 right minus 4. well now you're doing 25 plus 32. so now you'd say okay so i've got 5 plus or minus the square root of 25 plus 32 over 4. and this is a case where yeah there's not a lot of simplifying you're going to be able to do so your answers are that x is equal to 5 plus the square root of 57 over 4. and 5 minus the square root of 57 over 4. and so if we think of those as our values of a in the factor theorem that tells me that i could write 2x squared minus 5x minus 4. if i bring the 2 out front i could write that as 2 times x minus 5 plus root 57 over 4 x minus 5 minus root 57 over 4 and okay i factored my polynomial that's not where we want to go right no nobody likes doing this right occasionally you got to do it you got to use quadratic formula because there are solutions but they're gross that's probably not where you want to go you're hoping that you've got rational or even better integer roots for your polynomial so how do you how do you work backwards so let's say somebody gives you a polynomial like x squared minus five x plus six and they say okay i want you to factor this thing okay well one of the things we could do is we could fall back on on rational roots theorem rational roots theorem says that you know possible factors you know i could have plus or minus 1 plus or minus 2 plus or minus 3 plus or minus 6. so i could have x plus or minus any of those numbers and you can just try them and see if they work right you could use the you could use the factor theorem plug each of those numbers in see if you get 0 trial and error but another way to do it is to say well let's suppose that this you know let's say i could factor so let's say i can factor this as as x plus a times x plus b right or maybe you want to do x minus a x minus b okay so well then i can say well what do i get if i if i multiply that out so x times x x squared x times a x times b so i have a x plus b x i can write it like that and then a times b all right so if you compare the two well then you realize that there's two things that have to happen i need a times b to equal six i need a plus b to equal minus 5. so now you got to come up with two numbers two numbers that multiply to give you six and they add to give you minus five all right so possibilities are going to either one times six or two times three and well or minus two times minus three minus one times minus six right and of those possibilities the only one that adds up to give you minus five would be a is minus two b is minus three right or i guess this could be minus three that could be minus two that doesn't matter so then we can say oh yeah so that means that i've got x squared minus 5 x plus 6 and that's going to be equal to x minus 2 times x minus 3. all right that's that's your your sort of typical factoring scenario that you might be dealing with the only thing that might be a little bit more complicated is if you've got a coefficient in front of the x squared we're running a little long on time for this one so let's try to do this one quickly so let's say i have something like 2x squared plus three x plus one and i want to factor that okay okay so let's give this one a try what are we gonna do well one of the things that you might try is first factor out that coefficient right reduce it to a problem like the one that we just solved the only catch is now there are some fractions in there three halves plus one-half right which makes things maybe a little bit trickier um but we all we already know from the from the rational roots theorem that the only the only factors would be looking for could be plus or minus one or plus or minus one half right so we we kind of have things narrowed down a little bit so we think about okay we want to multiply to give one half add to give three halves one times a half gives me a half one plus a half gives me 3 halves all right so you do that kind of again you're thinking about solving these equations and so we say okay so x so 1 times 1 half right so we can factor like that if you if you don't like that one half in there you could always take this 2 if you want to you could take this 2 and you could put it back with that second factor and you could write it as x plus one times two x plus one if you don't like having the the one half uh the nice thing about having it this way though is now you know what the zeros are minus one and minus one half those are your roots so in our last video we looked at factoring quadratics right and of course among all the various factoring formulas that you know quadratic formula matrix and appearance right this is probably our most famous formula for factoring but there are a few other formulas that pop up from time to time okay one of the ones that comes up more often than you might think is the difference of squares formula the difference of squares formula says that if you have x squared minus a squared you can factor that as x minus oops a times x plus a okay that's your difference of squares let me clean up that a little bit there we go okay so difference of squares comes up fairly often sometimes just simply in factoring right so for example somebody gives you x squared minus 4 and you say oh yeah i know what that is it's x minus 2 times x plus 2. right or or maybe they give you something like x squared minus 3 and you're like well hang on a sec three's another square well true it's not a perfect square but it's the square of something right we're working over the real numbers we can have irrational values right 3 is the square of something it's the square of root three so we can factor that as x minus root three times x plus root three right so we can do difference of squares one of the other places where you might run into this is actually using it sort of in reverse to get rid of something like say a x minus root 3 or a root x so you might do it in something like say you have the following let's say you have x minus oh let's say [Music] 4 over root x minus 2. and you don't like having that root x in the bottom you want to get rid of it right so how do you how do you get rid of that root x well one of the things you can do is say hey you know i just i just saw this example here sure the root was on the 3 here rather than on the x but i have this difference of squares things going on so if i multiply by the the same thing but the opposite sign i'm going to square both of the terms right so if i take this and i multiply by root x plus 2 that's going to that's going to get rid of the the square root of course i can't do it on the bottom without also doing it on the top okay let's put extra parentheses in there just to make sure we don't mess it up and what do we get we get x minus 4 root x plus 2 and on the bottom root x times root x becomes simply x right and you'll notice that the whole point here is the cross terms cancel 2 times root x minus 2 times root x add those up you get 0 they disappear minus 2 times 2 x minus 4. all right probably at this point you're going to you're going to cancel those and simplify down to root x plus 2. okay good so we have that one note of caution a sum of squares is always irreducible so x squared plus a squared there's nothing you can do you can't factor this is irreducible um common mistake that a lot of students will make again one of these ones that's sort of born of wishful thinking is they they really want to be able to sort of factor things so they can cancel and simplify and there's going to be a sum of squares in there somewhere right x squared plus something positive you can't do anything about it okay you got you just got to leave it alone there's nothing you could do to simplify right because if this is a positive number you're adding a square we know that squares are never negative right so this can never be zero and if it can't be zero it can't have a factor okay now what about difference of cubes turns out you can do a difference of cubes and you can also do some of cubes okay so unlike sum of squares right you can't factor sum of squares you can factor a sum of cubes right the reason is that you can take this cube root of a negative number right if i was if i set this equal to 0 tried to solve i'd have a negative on the other side i can't take the square root i can take the cube cube root of a negative though so difference of cubes looks like the following x cubed minus a cubed is x minus a we know that's a factor because if i put in x equal to a i get 0. all right and the other term you square the first term then you multiply with the opposite sign a times x and then you square the last term a squared okay for sum of cubes same thing with the sign change this will be x plus a right now minus a is your root and this becomes minus ax right so you just interchange these two signs between sum of cubes and difference of cubes one important thing to to point out is that these these quadratic factors that you get from a difference or sum of cubes these are also always irreducible okay so if you're doing a factoring problem once you've applied a difference of cubes or sum of cubes formula you're done okay there's there's no further factoring that you can do you stop there okay so as a quick example somebody gives you something like 27 y cubed minus one over eight we say oh what do i know about those coefficients um 27 is three cubed one over eight is a half cubed okay so this is the difference of cubes so it's going to be so this is 3y and then i cube it so it's 3y minus a half now i square the first term 9 y squared opposite sign in the product so it's going to be plus 3 over 2 y and then plus a half squared so plus one quarter okay and you factored all right in this video we're going to quickly point out another strategy that sometimes works for factoring in this case for cubics this is probably something you're most likely to apply in the case of a cubic polynomial so what you do here when you're factoring by grouping is you kind of you pair the first two and the last two terms you kind of set those aside and in the first two terms there's a u that's common to both in fact there's a u squared that's common to both and so what you do is you factor that out so you say okay in those first two terms if i factor a u squared okay in fact i even factor out a 2u squared right there's a 2 that's common to both so i factored a 2u squared and i'm left with well here to get 2u cubed i have to take 2u squared and multiply by u right and here for 4u squared well i've got 2u squared i have to multiply by minus 2 to get minus 4u squared okay so you factor that out then you come to the last two and you say okay what can i factor out from the last two terms well there's a three that's common to both so i factor out the three okay and in this case i get kind of lucky because you'll notice that once i've done that i have the same factor here and here so think now about reversing sort of distributive property if you like i can pull that out as a common factor so i factor out the u minus 2 okay and then i'm left with 2 u squared plus 3. okay and that's as far as i go i can't factor any further right this is a sum of squares right 3 is the square of root 3. and and as we've pointed out a sum of squares is irreducible so you can't factor this any further you stop there if you're looking for roots there's exactly one when u is equal to 2 the polynomial is 0. okay so that's factoring by grouping it's it's something you can always consider but it doesn't always work okay we'll do a couple of examples in later videos where this fails and we have to look at other techniques i'll mention one more option before we move on to sort of some more general methods you might run into what you might call a quadratic in disguise and and these typically show up in degree four polynomials sometimes higher where you might have something like the following you might have say x to the 4 okay plus 3 x squared plus 2. okay so that's a degree 4 polynomial in general degree 4 polynomials they're quite difficult to factor they take a lot of work but one of the things that you might notice is that there are no odd powers in here there's only even powers right and in fact x to the 4 is really you know we could write that as x squared squared right and then 3 x squared plus 2. so if we wanted to we could think of this as like y squared plus 3y plus 2 where y equals x squared and we know to factor this this is going to be y plus one times y plus two ah but we should end in the variable we started with x squared plus 1 times x squared plus 2. right and because those are both sums of squares you can't factor any further both of those are irreducible quadratics right if we had gone with say a minus sign here we'd have minus signs there and then we could factor each of those as a difference of squares but that's not the example that we had so finally um with this um issue of trying to factor polynomials we'll come to long division so this is the most difficult of the various factoring techniques or at least the most time consuming it's also the most reliable this is you know if there's a method that's going to work every time it's going to be long division it's always going to do the job all right so here's a cubic polynomial that we might want to try to factor you might say hey grouping is pretty easy let's see what we can get away with if we do do grouping right but you'll notice right away that grouping is going to fail in this case if i take out an x squared um well i'm left with an x plus eight uh over here best i can do is a three right i could factor out a three and i'd be left with seven seven x plus nine it's not gonna work right doesn't so so grouping is not an option so we're like okay that's out what else are we gonna try well we can go to rational roots theorem right what are the possible factors here we could try um so possible integer roots well the possible integer roots are going to be the numbers that divide evenly into 18 plus or minus 1 plus or minus 2 plus or minus 3 plus or minus 6 plus or minus all right our last topic related to polynomials is solving inequalities involving polynomials this again is something that you're going to have to do fairly frequently once you get into calculus when you're trying to figure out things like where is the derivative positive where is the derivative negative this is going to be key to solving a lot of optimization problems curve sketching problems things like that so it's something that's going to come up fairly frequently we'll start simple we'll work our way up the first one is linear right so linear inequalities you learn going back to high school i'm not sure what grade you you see them in but certainly this is a basic thing that you would have seen in school and linear equality inequalities are fairly straightforward so here you can rely on the fact that well you can always add something to both sides of an inequality right so we can we can add minus 4 and we get so 4 minus 4 of course is 0. 3 minus 4 i get negative 1. so 2x bigger than negative 1 is equivalent all right i can divide by 2 and because 2 is positive i don't change the inequality and i'm done i've solved x has to be bigger than minus one-half right so i might also want to give an interval of solutions so we write it as minus one half to infinity okay so we solve that inequality now where where people get themselves into trouble is is you move from linear to things like quadratic and you want to apply similar techniques right and one of the things you might do so here notice you've got x's on both sides and you might have this tendency to say oh let's let's move the x over let's isolate the x's right it doesn't get you anywhere once you go past linear to quadratic cubic anything else other than linear there's really only one thing that works which is going to be to basically isolate for zero so you got to bring everything to one side okay so if i bring the 5x over and the minus 6 over this is the same thing as saying x squared minus five x plus six is bigger than zero okay so this is useful because now this this boils down to deciding where is the polynomial positive and where is the polynomial negative and we know how to solve that problem because we know that the only possible places where a polynomial can change from positive to negative are at the roots right so once we find the roots we know where the possible sign changes are and then we just have to determine signs on either side of the root in this case i mean we kind of know right what things look like here we can graph this it's a quadratic okay it's a quadratic opening up it's going to have a couple of roots right it's going to it's going to look something like this right they're going to be two roots so we know it's going to be positive on the outside negative in between right we also we know that just by looking at the leading coefficient right so either this is positive everywhere okay or it has an intercept one or two intercepts in which case is going to be positive outside the intercepts negative between we know that because the leading term has positive coefficients so we know that this is a quadratic that's opening upwards so how do we find those roots well we factor we've done this one before in fact we know this factors as x minus 2 times x minus 3. right so solving the original inequality amounts to figuring out where is x minus two times x minus three where is that bigger than zero so what you might do now is you might write down what we call a sine diagram so in a sine diagram you just draw yourself a little number line on that number line you mark off the roots so there are only two roots two three and then you put down the signs between each root so we know as we said they're going to be positive outside the roots negative in between right the other way to realize that is you can always you know you can always choose some test value like x equals 4 right if i plug in 4 i get 2 times 1 is positive so i know it's positive out here i know that when i cross 3 this x minus 3 factor right it's positive here but it's negative here so this is going to change sign this one hasn't yet so now i have one minus sign right and then once i cross 2 this one becomes negative as well now i have two negatives gives me that positive there are a few ways to work that out you could always do test values in each interval if you're not sure there's our sign diagram and we want to know where is this thing positive so we just look for plus signs right plus sign for positive so we know that our solution is going to be that x belongs to either everything from minus infinity up to two or from three to infinity okay all right one last one cubic inequality right so maybe there's a bit more work involved here but the initial strategy still the same bring everything over x cubed plus 2x squared minus x minus 2. we want that to be bigger than 0. okay so let's see if we can factor maybe we check to see can we actually group right because if we can factor by grouping that saves us from having to do long division that's always nice take out the x squared leaves me with x plus two here if i take out a minus sign ah i'm in luck x plus two that minus sign of course is just minus one okay factor one more time x squared minus one times x plus two we want that to be bigger than zero and that's a difference of squares x minus one x plus one okay so we need x minus one times x plus one times x plus two we want that to be bigger than zero so we draw our number line we mark off the three roots so there are roots at minus two minus 1 plus 1. and now we have to work out the sign in each interval so if we choose something bigger than 1 2 for example we can quickly see that all three factors are positive so the whole thing is positive if we choose something between minus one and one this first factor is going to become negative the other two remain positive one minus sign means the whole thing is negative between minus two and minus one now this factor and this factor they're both negative but that one's positive two negatives gives me a positive and then finally once we're less than minus two all three factors are negative three minus signs gives me an overall negative and that means that if i want this thing to be positive i look for the intervals with the plus signs i want x to be between 2 n minus 1 or from 1 to infinity right of course if this had been bigger than or equal to we could have done that the only difference is now we include the zeros and so the round brackets would become square brackets except on the infinity right we never include infinity okay in the next few videos we're going to look at rational expressions or or more generally well rational functions right so we've gone over polynomials we have a good idea of what a polynomial looks like a rational expression is just a ratio of two polynomials okay so we're looking at a ratio of polynomials so something that looks like say p of x over q of x okay so for example we might have something like x cubed plus x squared okay over 4x squared minus 8 x okay that's a rational expression okay if i if i were to assign a function to this so i'd have a rational function if i said f of x equals okay then it's a function so it's a ratio of two polynomials right so maybe we would we do want to think of this as a as a function the first thing we might ask for such a function is what's the domain right here's a function what's the domain of our function right now we have this convention in calculus that if the domain is not specified and it usually isn't then the domain of our function is just going to be the largest set of real numbers for which our function is defined um so for a rational expression rational function when is it defined well it's defined for all x for which q of x is not zero right that's the only real restriction here we can't divide by zero otherwise we know that polynomials are defined everywhere right they always give us a real number output and a ratio of two real numbers is always defined as long as the one on the bottom is not zero okay so we can do this so if we want to know where this thing is defined we've got to factor it okay and so up top we notice hey there's actually x squared it's common to both i can take x squared please move x plus 1. on the bottom there's a 4 that's common and x 4x times x minus 2 okay there's our fully factored version of this rational function now from here we want to say where is it defined right yes there's an x that can be cancelled and we're going to cancel it but we we really only want to cancel if it's non-zero right so if x is zero this is undefined this expression is undefined if x is zero because i get 0 over 0. so i don't want x to be 0. the other place where i don't want is i don't want x to be equal to 2 because if x is equal to 2 then x minus 2 is 0. okay so i have those two zeros in the denominator i want to avoid both of those now as long as i stay away from those two then yes i can simplify i can cancel this x with one of the two upstairs okay and i'm gonna get x times x plus one over four times x minus 2. all right there we go so now i've i've simplified right one of the things you want to be very careful with when you're simplifying rational expressions is that you can only simplify once you've factored don't try to cancel things if you haven't factored top and bottom right it would not be valid for me to cancel on x squared here with an x squared down here that doesn't work i can only cancel once i fully factored fully factored i can look to cancel but again i should keep track of the fact that i cancel that x because that does affect the domain for my function okay so you should keep track of that domain and you've simplified we're going to pause here we're going to come back we're going to look at some other information that we can get out of this thing once we've got it down to here all right so this is at least for now um our last video on rational expressions and functions um here we're going to look at rational inequalities right we looked at solving polynomial inequalities earlier and your first temptation when you see a rational inequality is to turn it into a polynomial inequality by just clearing denominators you say hey let's just multiply everything by x plus 2. then it's polynomial and i solve there's just one problem with doing that right we know we know from some of the examples we've looked at that we might have a sign change at x plus 2 right when x is equal to -2 there's a sign change we know that and and so that means that if we wanted to clear the denominator we'd have to do two separate cases when x is less than minus two and when x is bigger than minus two because when x is less than minus two you'd be multiplying everything by a negative number and we know that if you multiply by a negative that reverses the inequality okay so how do you proceed if you can't just cross multiply get rid of the denominators turned into a polynomial inequality well first step is the same as it was when we were doing polynomial inequalities we just you get everything on one side so we say okay so this is the same thing as saying x plus 6 over x plus 2 minus 3 less than or equal to zero okay now we're dealing with the problem of figuring out where is a function less than or equal to zero right and that's something that we know how to solve but the first thing we got to do is we've got to rewrite this function right because we want to get it in the form of a rational expression we want it in that form polynomial over polynomial and so we're adding up these terms one of them has a denominator so we know what we need we need a common denominator so these two terms that are missing that denominator we put it in x times x plus 2 over x plus 2 plus 6 over x plus 2. all right so we're going to do several review videos now on functions so functions are something that everyone sees in high school and you spend a lot of time on them and still it remains one of those areas where even at the end of a calculus course it can be clear that a lot of people still don't quite understand what a function is how a function works how do you manipulate functions how do you work with function notation these sorts of things it's something that can be a bit of a challenge for people so let's just start with some basic examples so functions you might have seen right the ones that are familiar from high school so you probably saw things written you know like this say f of x equals say three x minus two for example okay or g of x equals oh let's say x squared minus 4x plus 3 something like that right so these are your sorts of functions that you've probably encountered a number of times um you've seen this notation f of x uh in in subsequent videos where we're going to expand on this notation we're going to explain the meaning a little bit for now we'll just use it i'm hoping that it's somewhat familiar to most of you as you're watching this and you're probably used to using x as the variable in your function but there's no reason why you necessarily have to use x right you might have something like maybe velocity as a function of time something like that right so you might have something like uh you know minus nine point eight times t plus i don't know some initial speed something like that so these are all examples of functions that you've encountered right and some of these functions they have names right so special types of functions they have names to describe them all right this one here is an example of a linear function this is an example of a quadratic function and of course both of them are are special cases of the more general idea of a polynomial function but you can go well beyond polynomials we can look at at things like you know let's say we do something like f of theta equals sine 2 theta right some sort of trig function we might do something like that exponential functions logarithms we'll look at all of these as we as we proceed through the videos and of course the name linear here why do we call this a linear function well it's a linear function because if you were to graph it if you were to set y equal to f of x so y equals 3x minus 2. then of course we know that this this here is just a straight line hence linear right you see the word line in linear right there okay so we'll talk about graphs we'll talk about different types of functions we'll go over the function notation we're going to look at all these different elements of understanding functions and working with functions over the next several videos okay so i've put a definition for for what it means for something to be a function up here on the board this is uh let's say an informal definition of what it means to be a function but it works for our purposes in other courses you might see more careful more precise definitions of functions perhaps in a course on on discrete math or or a course on introduction to proofs or something like this you might see something more careful more precise but the the main ingredients in in what it means to be a function and and here our function is is f okay the main ingredients are well you need to have some sets okay so a b these are sets um what does a function do a function takes every element and we should probably underline the word each here a function has to go through every single element in the set a okay and it has to assign it to some element in the set b and one of the rules for what it means for a function to be a function is that any particular element of the set a can only be sent to one particular element in b so if you have a function where let's say let's say 4 is an element of your set you can't decide that you want to assign the number four to three different things you can only assign it to one particular value okay this is this is the defining property of a function if you relax this rule if you allow for something in the set a to be associated with more than one element in the set b you no longer have a function you have a more general object called the relation and relations again are interesting mathematical objects and something that you might study but it's unlikely that you will see that in your calculus course relations again or something you might see in a proofs course a discrete math course probably not in your calculus course now there are there are a few other things that we can add one of the ways that you might denote the fact that this function goes from a set a to a set b you might write this by saying i'll write f for the the letter that represents your function and a colon and then a arrow b so you'd read that as you know f goes from a to b or f is a function from a to b so there's there's this sort of dynamic point of view when you have a function that there's inputs that are being transformed into outputs right so in these basic examples that we had here right the x that you see here that's your input right that comes from some set a and over here is your output right so this this whole thing here that's going to be your output um now in these examples here these ones which are coming from sort of you know high school pre-calculus if you like chances are your input and your output are both real numbers and calculus inputs outputs they're always going to be real numbers right and and for functions like these any real number will do right this expression makes sense for any number x can be any number right it doesn't matter what that number is it always makes sense to multiply it by three whatever that output happens to be it always makes sense to subtract two right any number it can go right the rule in this case is this instruction right a lot of people think of the formula as the rule right but really the formula there is the output the rule is this instruction saying that whatever the input was you should multiply by 3 and then subtract 2 right but we don't want to have to actually say that every time this is why we develop mathematical notation it's a lot simpler to just write down the formula than it is to express that rule in words okay a few other bits of terminology before we move on the set a so the set from which the inputs come this is known as the domain for your function the set b is known as the codomain okay another way that you might that you might represent this function process is you might have a little diagram that's something like this here's my a here's my b and again we draw an arrow to represent the fact that things are coming from a and going to b and if we want to say that f is the rule that's that's doing that we can write something like that um so this is another notation that you might see unfortunately in calculus we tend to kind of gloss over some of this we don't necessarily use this notation we usually don't specify the sets a and b because those sets are always going to be subsets of the real numbers and so in calculus we get into this habit of letting the formula define the function right but the formula is really only part of the definition the domain the codomain those are parts of the definition this is something that will come up later when we talk about inverse functions right when we talk about inverses we need functions to be one to one something like like this function here it's not one to one not everywhere but if i if i shrunk the domain if i went with a smaller domain if i didn't use all real numbers i could shrink this down to something that is a one-to-one function right we know that the graph of this is a parabola if you just took half the parabola you'd have a one-to-one function so there are there are concerns like that that do come up from time to time in calculus but most of the time we're a little bit lazy about this and again we'll we'll talk in another video about how we make sense of domain in the context of calculus where we never explicitly write down these sets a and b all right we're going to look at a few simple examples of functions in this video just to kind of expand on this definition get a little bit better feel for for what this definition is saying what does it mean for something to be a function so here's an example and here's one we're just going to explicitly construct a function so we're going to say a a is going to be the set containing the letters 1 2 three b it's going to be the set containing letters let's say a and b and i'm just going to define my function like this and there's a few ways that we could do this so one of the ways we can do it is we could just visually represent this by saying okay here's a here's b okay so we just kind of draw those two sets and we're going to mark off one two three we're going to mark off a b and we're going to specify the function by just saying you know what goes where right so we could do something like this we could send one to a okay you do that i could send two maybe two goes to b okay we could do something like that now if i stop here i haven't actually defined a function because one of the conditions for a function is that every element of the domain has to be assigned so until i send this 3 somewhere i don't have a function so maybe we send 3 maybe we send that to a as well ok so we have we have a function right now you might be concerned that there are two different elements of the domain that both get sent to a this is okay all right you're allowed to have this in a function what you can't have is if i had one if i had two different arrows coming out so if one went to both a and b i would not have a function so for any element in the domain you can only have one outgoing arrow but in the codomain you can have more than one incoming arrow right so it's okay that i have two arrows going into a it would only be a problem if i had two arrows coming out from one of these elements okay so this has a function right this or sorry this defines a function this property if you are concerned about this this is where this this notion of a function being one to one comes in and we might talk about that later on right um now another notation you're probably familiar with is that if you have an element a assigned to an element b then usually what you do is you would write this you would express this by saying that b is equal to f of a right so over here in this context one of the ways i could define my function i can completely define my function by just again saying what happens to every input right once every possible input has been assigned to an output my function is defined so i could say f of 1 equals a f of 2 equals b f of 3 equals a so if i simply gave you that information that again would be enough to define a function okay this is fine all right now of course a lot of the time we're dealing with examples like the ones that we had up here a second ago where a is going to be the set of all real numbers maybe b is well it could let's let's say b is going to be the numbers from 0 to infinity could be all real numbers and and let's say i define f from a to b is going to be defined [Music] the following formula we say simply f of x equals x squared right so this notation right is being used here so what it's saying is that x in this case this x right this is my input it's what i called a over there all right so this is an element of r of a right this x squared that's my output so that's an element of b right we know that if you square a real number the result is never negative so this is indeed an element of that set b right okay so again that's another example of a function but you know there are many more examples we could go with something like like the following we could say so we might write down so here's something you might see in like a a third or fourth course in calculus you might see something like f of x y equals x squared plus y squared right so you have it looks like you have two inputs right there's an x and a y um this is in in calculus we usually refer to this as a function of several variables um but we can always think of the input here as the ordered pair which is an element of the plane of r2 right and the output would be this number x squared plus y squared which is an element of r right um of course it's also an element you know again this is not never negative so we could go with zero to infinity if you want there's no requirement though that your function hits every element of the codomain right so you can always make you can talk about the distinction between range versus codomain right the range is the set of all outputs that your function has right so here we might say well that's so the codomain is r but the range is the set of real numbers from 0 to infinity you can make this distinction between range and codomain when the range equals the codomain there's a word for that you can talk about a function being onto onto functions are usually not discussed in calculus the only place where you might have to think about it a little bit is when you talk about inverse functions and even there you can usually kind of gloss around generally what you do is you just kind of assume that the codomain is the same as the range this is something you can usually get away with in calculus okay so there's a few simple examples of functions we'll look at some more calculus focused ones in later videos but you know just just to press home this fact that or this idea that you know this definition of function it's very broad you can look at all kinds of different scenarios you could look at something where maybe maybe you take a and b to be something like uh i don't know the set of all let's say female humans and the relationship is is mother to daughter and you could ask is that a function well that's not necessarily going to be a function because there are some mothers who have more than one daughter right so you might have um two well i don't know it seems crude to call this an output now but two outputs for a given input on the other hand um any given daughter has only one biological mother um i guess well you know i don't know these days so so you could go in the other way and say that is indeed a function going in the other direction so there's lots of different scenarios like that that you could consider um but we'll typically be looking at you know objects like this most of the time when we talk about functions all right continue with our discussion of functions we'll see a little bit more about notation okay so a lot of people i think will get tripped up a little bit on the notation sometimes get mixed up on necessarily what are you actually meaning when you write down something so let's say you write down something like f of x equals 3x squared minus 2x plus 1 right you write that down what is what does that actually mean what are you what are you telling me when you write down an expression like this well again you haven't quite necessarily told me everything there is to tell me about this function because you haven't said anything about domain or codomain you've just written down a formula right but this this x that's in brackets here right this is this is your variable right so this is your it's your input or we might think of this as a as a variable um if you've done any let's say computer programming or something like this um you might have encountered this sort of scenario right where you have various functions that are defined in your in your program and they take some input right and probably you even put that input in parentheses like this and you run the function right you run the program and something happens to it so that's exactly what you're doing here right you're you're thinking of well you're going to take a program you're going to feed it some input it's going to do something to that input it's going to give you an output that's that's what's going on when you're working with the function okay so if we wanted to be more precise we could say something like you know we could say well f it's a function from well in this case maybe we want to say that it's defined for any real number right goes from r to r right and if we want to talk about this relationship between inputs and outputs we could say something you know like we could say that a you know we might write something like this a goes to b we might say b is equal to f of a um we might even write down our form and say well what is f of a well f of a is and again the fact that i'm using a different letter doesn't matter it's 3a squared minus 2a plus 1. right there's nothing special about x you can use whatever letter you feel like using right you want to use a use a you want to use x use x just make sure it's clear from the context what you're talking about and of course just like in a computer program you can take that variable you can substitute it for a value see what happens so we might say something like well what what do i get if i do f of two okay well f of two we're just going to replace right you look at the formula you say okay well on this side right in this expression i've taken a i've replaced it by two so what i do is i look for everywhere i see an a and i replace it by two three times two squared minus two times two plus 1. and if you're so inclined you can do that calculation get an answer in this case happens to be 9. right and doesn't matter what number you choose let's say i want to do f of minus one i can do that as well right three times minus one squared subtract two times minus one plus 1. okay notice i'm using parentheses around my input into the function this is good practice especially once you get into negatives because you don't want to miss something like the square of a negative right you don't want to make sign errors because you're being careless so minus 1 if i square that i get plus 1. so this is positive 3 right minus 2 times minus 1 again right double negative becomes positive so 3 plus 2 plus one i get an outcome of six okay that's all well and good what about what if i asked you for something like um f of y all right this really throws people off right because we're so used to writing y as equal to f of x when you talk about graphs right when you want to graph a function and we'll talk about graphs in simulator videos you set y equal to f of x where x and y are the coordinates in your cartesian plane and and you plot your function but here y is just some other variable name so what's f of y f of y i haven't told you anything about y so all i can really tell you right now is that f of y is three y squared minus two y plus one right that's all you can do and that's that's the correct answer if i now went ahead and told you something about y if i said oh by the way y is equal to f of x or y is equal to x plus 3 or something like that right if i told you that y had some particular value well then and again this is the whole point of function notation you can plug that value in but if you do it on one side you must do it on the other side so if i told you for example that y was equal to say something like x plus h i could say well what is what is f of x plus h that's something you'll be calculating quite a bit and and this is something where a lot of people will get messed up right when these to the plus h x plus h um some of them are one we're gonna want to do f of x plus f of h um some people are gonna just take f x and put a plus h on the end none of these are going to work out correctly because what function notation is telling you to do is it's telling you to take your input right and you and this tells you what should happen to the input you should square the input multiply by 3 then you should take the input multiply by minus 2. then you should add those two together add one more and you get the result so when you say f of x plus h what you mean is 3 times x plus h whole thing squared minus 2 times x plus h again leave it in parentheses plus 1. okay and from there you can if you're so if you're so inclined you can expand right um we know how to do foil in fact we might even remember the the formula for the square of a binomial from a previous video write that out we can put the constants through the brackets 3x squared 6xh 3h squared minus 2x minus 2h plus 1. right expand it out there's not much more you can do with that you could try to group things together or something but yeah pretty much you leave it at that um so the key is remember that when you're when you're working with functions when you have expressions like this right this kind of notation that's common in calculus right remember this is an instruction think of it as running a program it's telling you what to do with an input right so whatever the input happens to be you need to take your variable as it appears in the expression replace it everywhere by that input simplify if necessary to figure out what your output is going to be okay so we've mentioned now a few times that in calculus we have these conventions around domains right we we don't usually specify domain and codomain when we write down a function generally what we do is we just write down a formula like this one here right and and so in a sense we're being a little bit careless when we do that uh but we we get ourselves out of trouble because in calculus there is there's a convention that we're all working with so we all agree on domain okay and so this the assumption that you make is that your domain is the set of all real numbers unless well there's a couple of scenarios one is you might be in some actual practical applied context where it doesn't make sense for a function to be defined for all real numbers right um you might be doing some sort of applied problem maybe you're trying to optimize a length let's say right so we know that length is a quantity which can't be negative it doesn't make sense to talk about a length of like -5 meters so we wouldn't talk about negative lengths so we would be in a context where we're only looking at positive numbers for input right so sometimes there is a domain like that that's that's stated you have this applied domain which which you deal with but most of the time the domain is left implicit and if the domain is not specified then you check to see if there are any values where your function is undefined right so if you've got something like a polynomial function like this quadratic it's defined for every possible value of x so the domain is indeed truly r it's all real numbers but maybe you're dealing with like a rational function right maybe you're dealing with something like 1 over x squared minus 3x plus 2. and you realize oh wait i can factor that denominator right and i can factor that as x minus 1 times x minus 2. so in this case if i wanted to specify the domain and one of the notations that people will use for that is dom for domain of f um and if you like you can think of this as as itself it's it's a type of function right it's a function that takes a function as an input and gives you a set as an output that set being the domain okay so the domain of f is well there's two ways you could do this um you could use this set builder notation and say well it's a set of all real numbers x such that well what do we have to avoid we have to avoid dividing by zero so i can't divide by zero um so that means x can't be one because one minus one would be zero x can't be two because two minus two is zero right so there are two values of x that are not allowed so i could write it like that if you want you could also write it as an interval you could say well it's it's everything from minus infinity up to one open bracket because we don't want to include one union everything from one to two again open brackets and then everything from two to infinity right that's another way that you could write that set another example we could do something like this we could do let's say g of x is the square root of 4 minus x squared okay now again we're working over the real numbers when we're doing calculus we're not looking at scenarios where we might allow for complex numbers imaginary numbers are not part of the equation here we know that if you want to stick to real numbers you can't take the square root of a negative right because we know that for any real number if you square it the output is going to be zero if you're squaring zero otherwise the output is positive right so you know if you think about what a square root is doing a square is asking you know which real number when squared will give me this result and of course it might be that there are no numbers so we ask ourselves okay we want the square root to make sense well that means you need 4 minus x squared to be bigger than or equal to 0 right you want it to be positive maybe you like having the x squared out front we've looked at inequalities already if you if you've switched the order right that's you're multiplying both of these by minus one right you're flipping this sign remember that when you change the sign the inequality reverses this can factor x minus 2 x plus 2 we want it to be less than or equal to zero so we draw ourselves our little number line we know that there are two places where this is equal to zero it's equal to zero at two and minus two and in each of these three intervals that results we can do test values and we can check and we find that it's positive here it's positive here and it's negative in between and we want the outcome to be less than or equal to zero so that means we're looking for the minus sign right we want it to be negative so the domain of g in this case so again we could write it in set notation set of all real numbers x such that x is between -2 and 2 but of course that can also be easily written as a closed interval from minus two to two okay um so this is typically how we handle domain and calculus right somebody gives you an expression like this the first thing you want to ask yourself is okay um are there any values of x are there any real numbers for which the function is going to be undefined right are there any inputs that will not produce an output i want to know what those are because i don't want to consider them they're not going to be part of my domain all right so we're going to look briefly at graphs in the next couple of videos later on once we do get to calculus we will be looking at techniques for producing graphs of functions the nice thing about graphs it gives you a very concrete visual way of understanding what's going on with the function right sometimes when we write down something like you know even something as simple as a quadratic you know we write down this sort of expression so y equals f of x right this is uh this is the sort of equation that you see which usually signifies we're dealing with graphs and and let's say that function is something like let's say it's a quadratic right x squared minus 3x plus 2 something like that so just looking at this function maybe maybe you can't immediately tell me everything there is to know about that function right um we might want to know things like you know which over which intervals is the function sort of you know increasing you know getting bigger as x goes up where is it decreasing where is it going down right um so we want to know things like that we want to know you know is is there a place where it kind of bottoms out you know is it does it reach a maximum there there are lots of things like this that we might want to know about a function right and all this behavior is relevant right because we're using these functions to model things we want to you know model let's say phenomena that vary over time right so we might be interested in things like maybe even some of the value of a stock we want to know is it increasing over time is it decreasing over time probably it's increasing some weeks decreasing other weeks right things tend to fluctuate go up and down we want to know what's happening and sometimes the easiest way to see what's going on is to look at a graph now somehow as soon as you put this y in there there's this understanding that okay if you just have f of x you know in the formula now we're talking about a function right but but if i instead of putting f of x i put y oh now it's a graph so so why do we have this this context why do we um as soon as there's a y in there we're talking about graphs well really what the graph is and we can talk about this for any function let's say f from from a to b really what it is is it's the set of all ordered pairs a comma b so this belongs to a set called the cartesian product you may not have seen this notation before but don't worry about it because it's not going to come up that often but this is just a way of denoting the fact that we want a to be an element of a we want b to be an element of b so the set of all ordered pairs where the first element in that pair comes from a the second one comes from b with the property that b is the element of b of big b that little a is assigned to by f right so it's instead of all b such that b equals f of a so this is this is where the idea of a graph comes in um and and the significance of x and y here is that these are the default variables when we're talking about things that live in what's often written as r2 so r times r right so elements here are of the form x comma y where x and y are real numbers and the important thing here is that this is an ordered pair so there is this notion of first and last right y comma x is not the same thing as x comma y when you're talking about ordered pairs and we have this visualization right so we have this cartesian plane okay so this is an idea that goes all the way back to rene descartes and it was actually quite a revolutionary idea at the time so before descartes geometry and algebra were very much different topics they were somehow not unrelated but you know people tended to kind of do one or the other we they didn't really see connections between the two and they can't realize that algebraic expressions like this like setting y equal to f of x could be visualized could be viewed on a graph and the cartesian plane this cartesian coordinate system is this grid system which we all know and love right we draw a pair of axes we label the horizontal axis as the x-axis we label the vertical axis as the y-axis and the the the big revelation the big idea the big realization that descartes had and um the the story is that he had this revelation while lying in bed watching a fly crawl along his ceiling and he realized that he could describe the location of that fly if only he knew the distance from that fly to two of the four walls in his room right so we would have x being the distance measured from one side and y being the distance measured from the other side and typically we would mark these distances on the respective axes so we would mark x here and we would mark y there right so so once you have this idea of the cartesian plane well now you can visualize a function you can visualize the graph you can plot the graph because for every ordered pair for every xy that satisfies this equation you can plot a point right and and so the the most elementary thing that you can do here is you can just start choosing x values and seeing what the corresponding y value is right so we could we could go through and say okay when x equals zero let's see if x equals zero y is equal to two right so i kind of mark off you know one two and i plot a point right when x is equal to 1 i would have 1 minus 3 plus 2 i get 1 right so when x is equal to 1 y is equal to 1 and i plot a point right and so on actually sorry that's not even correct is it when x is equal to 1 y is equal to zero all right when x is equal to two four minus six plus two oh when x is equal to two again um y is equal to zero okay uh when x is equal to three you can work that out y is gonna be equal to two again and you know but then you say well you know it's not gonna be this like you're not gonna draw straight lines in there you want to you want to get a better idea of the shape so you start filling in more points you might say well what happens at 1.5 you find out that there's a y value down here and so on and eventually you can fill in all the points all right and you produce your graph you get this this graph of a parabola right and you can't individually plot every point on the graph because of course there are infinitely many of them and and and you know it becomes pretty inefficient to use this technique of just plotting points right so you want to develop techniques you want to come up with with methods for quickly producing these graphs so that you can use that graph to understand the function you don't want to sit there generating point after point after point after point well unless you're a computer right if you're a computer you can quickly generate enough points you know within the sort of pixel density of your screen to produce something that looks good to the human eye but it's not a very practical process for humans so we want to look at some other options most of those options are going to come later on once we get into into calculus right in calculus we're going to learn techniques for understanding graphs of functions for now we're going to look at a few basic examples and a few basic principles to understand graphs of some simple functions all right so in the next few videos we're going to look at function arithmetic so this is the various ways that you can combine old functions to get new functions okay so basic operations are addition and subtraction okay and they're defined exactly the way you think they should be defined so if i have two functions f and g and i want to add them together well as usual in calculus if we want to specify a function we just have to tell you what it does with the given input so f plus g is the function that if you give it an input x it's going to calculate f of x it's going to calculate g of x and then it's going to add them together simple enough right and subtraction it's the exact same story right if i wanted to do f minus g i do f of x minus g of x okay so that's pretty simple so for example if if f of x was something like the square root of x and g of x is something like 1 over x minus 2. and i want to calculate f plus g well it's just f of x plus g of x so it's root x plus 1 over x minus 2. simple enough right one thing that you should probably watch out for when you're doing these is there is there is a domain issue to be aware of right um so the domain of f plus g well in order for the right-hand side to be defined right root x has to be defined 1 over x minus 2 has to be defined or in general right to define the left-hand side you need to be able to find the right-hand side the right-hand side is the sum or difference of two numbers so both of those numbers have to be defined so that means that x has to be in the domain of f and it has to be in the domain of g right it has to belong to the domain of both that means that we need to take the original domains and intersect them okay so for the example that we have on the go here we know that f x has domain 0 to infinity g has domain well everything but 2 right so minus infinity to 2 union 2 to infinity so if we intersect those two domains what happens is well we basically have to remove 2 from the domain of f right so f plus g is going to be defined if x is bigger than or equal to zero but not equal to two so the domain for f plus g will be from zero to two and from two to infinity okay all right so that's simple enough uh from here you could move on to looking at multiplication [Music] okay so multiplication you define more or less the way you think you should right [Music] the product fg evaluated at x is just the product f of x times g of x okay and and the same domain rule applies here as as we had for addition and and of course this would also apply for subtraction right i want to multiply these two numbers so i have to make sure that f and g are both defined fair enough so if we were going to do our example here if i did f times g at x i'm going to take this i'm going to multiply by root x probably we'd combine that as a single fraction and write it like so okay and you'll notice there's the same domain issue that you had before okay the last one is division of these basic operations and as you might expect f over g evaluated at x is just going to be f of x divided by g of x but here we also there's an additional domain restriction of course you can't divide by zero so g of x needs to be non-zero okay all right um so if maybe this function g of x you know this we could think of this as having come from the constant function one and the linear function x minus two right both of those have domain all real numbers but when i divide those two functions the one in the denominator has a zero and so that zero needs to be removed from the domain right so 2 is not part of the domain curiously enough this is an interesting one if i if i use our example here and this is probably a good thing to mention uh in the case where f of x is root x and g of x is one over over x minus 2. so if you do f over g at x so that's root x divided by 1 over x minus 2. right and when we divide we multiply by the reciprocal so we get root x times x minus 2. you might be tempted in this case to say that the domain now includes two the domain is just zero to infinity because you're multiplying by x minus two instead of dividing by x minus two um that's not the case however even though this this simplified form here is defined when x equals 2 the way we arrived at this was we started with this function g of x and we divided by it right and g of x is undefined when x is equal to 2. right so even though the final expression does appear to be defined at two if we're arriving at it through this division process well you can't divide by a number that's undefined okay so the domain there would still would still be the same domain that we encountered for addition or subtraction or multiplication okay so we're going to look at one other way that we have for combining functions which is function composition now function composition is is is fairly complicated and in some ways more complicated than it at first seems um so the the basic idea which i'm sure you've probably seen is well if somebody hands you two functions f and g you define this new composite function f o g whose value at x is given by first evaluating g at x and then taking that output and using it as an input for f right so g is evaluated at x f is evaluated at g of x okay sometimes it helps to sort of think schematically about what's what's going on when you're doing a function composition so if if g is a function from a to b okay so then that means that we start with a right that's where x lives x is in here and we apply this function g now we're over at b right that's where g of x lives now i want to use g of x as an input for f that means that f well f needs to have b as its domain okay f should go from b to c so then i can define f here okay and f of g of x will make sense and so the composition is really the one that kind of takes you from here to there f composed with g so you can give it a sort of the direct root that doesn't pass through b that's the composition now occasionally we can relax things a little bit here f doesn't necessarily need to be defined on the entire co domain for g but at minimum it has to be defined on the range right so one of the things you need right sort of a necessary condition here is going to be that the range the range of g has to be a if you like a subset of the domain for f right so every output for g has to be an allowed input for f that's what this is saying right in order to define the composition so those are the basic ground rules right so as far as when is composition defined if i wanted to write down the domain um so the domain is going to be so if we're working over the reals it's going to be all real numbers x that one belong to the domain of g and 2 when plugged into g give me something that belongs to the domain of f right so you kind of have to think carefully about how to form these compositions if we want to look at some basic examples we could look at say f of x equals x squared we could look at g of x equals oh let's go with something like an exponential function okay so we can do that so then it makes sense to ask well what is what is f of g of x well we can kind of think about it two ways we can think about this as f of g of x so f of e to the x so that would be taking e to the x and squaring it okay or i guess you could also think of it as you could think of it as you could kind of here i've kind of told you ahead of time what g of x is but not yet what f of x is we could do it the other way around we could also say that it's g of x squared but then of course once you plug in that g of x is equal to e to the x you're back at the same spot right now you might do one more step if you remember your laws of exponents if you square a power right a power raised to a power you multiply the exponents so you get e to the 2x now one of the things that's worth pointing out here is that order of composition is is important right so going going from a to b to c is not the same thing as as if you did f first and then g you'd be going from b to c and then trying to get to a well i guess you'd have to hope that c and a have something to do with each other right now of course we're we're in this situation where a b c all these sets they're always subsets of the real numbers so generally we're okay in this case both of these functions are defined for every real number so we don't have to worry about domain we just have to worry about putting these things together so in this case if i'm doing g of f of x well this time i'm taking g of x which is e e to the power f of x okay where f of x is x squared so what i get is e to the x squared okay and that's a very different result from e to the 2x those are not the same function okay so order matters um if we do one more example let's let's keep the same f of x let's take uh let's take h of x to be the square root function okay so now we could ask what is f of h of x okay so f of h of x would be well inside function is the root function outside function is the squaring function square root of x squared right and and of course if you square a square root square root goes away you're left simply with x um now you have to be a little bit careful you don't necessarily want to leave it just at that because if we do want to pay attention to domain right in order to define this composition right i first need x to be a valid input for h and i have to make sure that h of x is a valid input for f now everything is valid input for f but h of x does not take negative numbers as inputs so i have to actually put a little condition here that f of h of x is equal to x but i can only consider x bigger than or equal to zero okay so you have to be a little bit careful about that if you do the other order of composition there's also something interesting that happens h of f of x so now you're going to take x squared and you're going to take the square root and again you might be tempted to say that the square and the square root should cancel and just leave you with x well once again that's actually only valid if x is bigger than or equal to zero so if you think about what happens when you put a negative number in there if i put in some like minus 2 i square it minus 2 if i square it i get plus 4. if i take the square root of 4 i get 2 i get positive 2 not negative 2. so any any negative number that i use as an input here will be made positive all right same number opposite sign well we actually we know a function that does that this is actually the absolute value function so those are some basic examples using function composition um you can you can look for more of course choose any functions you like you can always plug one function into another here's uh here's maybe one you could try as an exercise so we could take something like uh f of x is x squared minus two x g of x is going to be let's say x over x minus 1. and you can try doing composition in other order right the important thing to remember here is keep a good handle on function notation if i'm doing f of g of x right that means i take g of x and everywhere i see an x i plug in g of x all right so f of g of x we could write that as well g of x squared minus 2 times g of x all right your next step would be of course to plug in g of x and then if you want you can simplify and then you could try the other order if you want right but you got to make sure you slow down do these things carefully it's easy to make algebra mistakes if you're if you're not being careful with these all right before we move on from function composition let's say a few quick words about inverses we'll come back to inverses later on in calculus once we're looking at derivatives of inverse functions and things like that but now is probably a good time to at least cover the basics right so if i have a function f going from say a to b right so i have some element x here and x gets sent to some y right so y is f of x the inverse i want the inverse to go the other way all right so the inverse should be something which undoes what f did and so that means that uh the inverse which we usually denote with this minus one superscript maybe we should just call it g it should go the other way it should go from b to a okay and if i start with f of x i should get x okay simple enough so what that amounts to saying is that if i do f inverse of f x i should get x and similarly if i did f of f inverse of well maybe we should call it something else maybe we should call it uh what do we call it y all right y was f of x all right so if i if i start with this element y here i apply f inverse right that gives me x okay so so i apply f inverse to y right remember this is y so f inverse of y is equal to x but f of x well f of x was y right so when you combine a function in its inverse whichever order you choose to compose them in notice that you end up where you started okay so there's a there's a name for this there's something called the identity function so maybe we call it say i right i of x equals x right every set comes with an identity function that just associates every element with itself right it goes from a set to itself and so what this amounts to saying is that f composed with f inverse is the identity function if you like this is the identity function on on b and f inverse composed with f is identity function on a right so they cancel each other out so this is where the idea of the inverse comes from right they cancel each other out in the same way that a number and it's negative cancel each other out if you're doing addition right they cancel and leave you with zero zero is is sort of an identity element for addition right because if you add zero nothing happens in multiplication a number a non-zero number in its reciprocal are viewed as inverses of each other because if you multiply them you get one and if you multiply by one nothing happens right a function in its inverse right these are you know again you use this word inverse this time it's with respect to composition because if you compose them you get the identity function and if you apply the identity function nothing happens okay so that's where that's where this notion of inverse comes from that's where the word comes from but the the caution here is that if you want this thing to be a function i think orange is done your function f must be what's called one two one okay um so there's this this property of being one to one and and usually the way you characterize this is you say well if f of x one equals f of x 2 for sum for some numbers x 1 and x 2 in the domain well the only way that can happen is if x1 and x2 were really the the same thing another way of putting this is that if x1 isn't equal to x2 then f of x1 can't equal f of x2 right so this is saying that you can't get the same output for two different inputs right for a regular function that's allowed you know for functions are allowed to have this happen right you can have more than one input give you the same output but if you want to if you want to reverse things or if you want to undo if there were two different elements of the domain they get sent to the same thing and you're sitting there and you're staring at this y in b and you want to get back to x right well if this y came from two different x values how are you going to choose which one it should be right you don't actually have a function right it's not a function if this if this input for the inverse could be associated to more than one output right so there's this issue of being one to one that you have to check we can we can do one example with this to show you how you check for a function being one to one how do you find the inverse um and then we're going to move on to uh two other types of functions all right so let's look at some simple examples of one-to-one functions and they're inverses okay so remember that one two one means that for any possible inputs let's say x1 and x2 if f of x1 equals f of x2 then x1 equals x2 and this is a fancy way of saying that you can't have the same input for different outputs right so if i have two outputs or sorry same output for different inputs right if i have two outputs that are the same then they came from the same input right so if we do something like let's say let's do something simple like a linear function 3 x minus 2. now you can almost work out what the inverse should be if you just think about if you think about the function in the sense of it's a rule that tells you how to do something right so what does this function tell you to do it tells you to take a number multiply by 3 and then subtract 2. so if you were trying to undo that function just like the undo key if you're working on a word processor or something like that right you always undo in the opposite order right you undo the most recent thing first so first thing i'd have to do it is i'd have to get rid of the minus two so i'd have to add two that would cancel out the fact that i subtracted 2. then to cancel with the fact that i multiplied by 3 i'd have to divide by 3. all right so that tells me that i should be able to reverse this function the way i confirm is you know we say well let's suppose let's suppose that f of x1 does in fact equal f of x2 for some x1 x2 well then 3x1 minus 2 would have to equal 3 x 2 minus 2 and if i add 2 to both sides 3x1 would equal 3x2 and i can divide both sides by 3 and i confirm that indeed x1 is equal to x2 all right now how do you how do you go about finding that inverse okay well one of the things that you'll notice here is that one of the things that's kind of hiding in this is that this is x right so if if y is equal to f of x then x is f inverse of y so what i should really do is let y equal f of x which is three x minus two then y plus 2 is equal to 3x and that means that x is equal to y plus 2 over 3 exactly as we said to reverse this function we should first add 2 and then divide the result by 3 right so that means that f inverse of y is y plus 2 over 3. and if you want to write that as a function of x rather than a function of of y that's fine remember that the the y is just some dummy variable it's a placeholder so if you want to put an x here just put an x there and you've got it right um you can do it that way okay i'll give you one more example let's go with x minus 1 over 2x plus 3. okay i'm going to leave it as an exercise to confirm that this is indeed a one-to-one function uh it's not so bad there's a little bit of work involved you got to cross multiply is doable so if we want to find the inverse let's set that equal to y right set it equal to y y is f of x f inverse of y should be x so we've got to take this equation solve for x so let's see let's cross multiply x minus 1 is y times 2x plus 3 which is 2xy plus 3y okay now let's think for a second what are we trying to solve for we're trying to solve for x so let's get everything with an x on one side x minus 2 x y is equal to 3 y plus 1. okay so i want to solve for x next thing i should do is i should factor an x out from this left-hand side x times 1 minus 2y is 3y plus 1. and so if i want to solve for x just have to divide by 1 minus 2 y so f inverse of y which is x is 3y plus 1 over 1 minus 2 y and there you have it all right so in the next couple videos we're going to take a look at exponential functions now earlier in the algebra review we already looked at laws of exponents we went over some of the basics there so you'll you'll remember that you know for um well if k is is a natural number so if k is 1 2 3 and so on when we write a to the power k we just mean a times a times a right it's we think of this as repeated multiplication so it's a times a times a and you do that k times and and that works for a while but but you know eventually you want to generalize to cases where maybe k is an integer so we allow for exponent 0 we allow for negative exponents and then you want to move on to maybe allow for rational exponents as well and so eventually you know you you come to the you know your rules right so we have these rules and i'm not going to repeat all the laws of exponents we've seen some of them but we know that for example a to the let's say m plus n is the same thing as a to the m times a to the n right and we say that a to the minus k is the same thing as 1 over over a to the k and you can you can make sense of these in terms of this this basic idea that exponentiation is repeated multiplication because you know if you think about division right if you're if you're dividing right so for example let's say i have something like 2 to the 4 and i want to multiply by 2 to the minus 2 right well on the one hand this rule here says that that should be 2 to the 4 minus 2 so it should be 2 squared which is 2 times 2. on the other hand if i'm doing 2 to the 4 times 1 over over 2 squared right then that's well it's 2 times 2 times 2 times 2 over 2 times 2. all right and 4 on the top 2 on the bottom you can cancel there's 2 left over again you get 2 squared and so on right you can make sense of something like you know a to the 1 over n why is that equal to the nth root of a well that's because if i do a to the one over n to the power n right well that's a to the one over n times a to the one over n all right and again we do that n times so it's a to the one over n plus one over n all right again 1 over n added to itself n times is a to the n times 1 over n which is just a to the one right you get a so so you can make sense of all these rules um and and basically you know you start with this idea of repeated multiplication from repeated multiplication you derive these rules and then at some point you want to move on right you want to consider negative exponents you want to consider rational exponents we want to be able to define exponential functions as a function of a real variable right so if you want to define things as a function of a real variable well then you kind of have to generalize it and the way you generalize is you say well really ultimately you just say that the rules the rules the thing right so we have this one the other one that's missing which we already sort of see in action there is is this one right so you take these rules and you kind of generalize and you use this to produce a function right and so for any positive real number a all right so a here is a real number you can define a function so you define let's say f of x equals a to the x all right um so in this case right the your input x is a real number and and you can actually do this for any real number as long as a is positive this will be defined for all real numbers x and the output is just a raised to that power and so you say okay well do we know what we're doing here do we know how this works well we we know what to do if a is sort of x right so we know this for we know what this means if if x is a natural number right if x is a natural number then we have this repeated multiplication okay we know what it means if x is an integer right because we know how to handle negative exponents and and we know how to handle a to the zero right because if uh if if n is equal to minus m let's say m minus n right i get zero right but then you're just doing a number divided by itself you get one so we know that a to the zero should be one right um and and so then you say what about if we have a rational number what if x is rational well actually we know how to handle that too because we know how to deal with reciprocals right things where you have one over over an integer we know how to deal with that and we have this rule here so if i wanted to do a to some rational number let's say something of the form p over q we know that we can do that as a to the 1 over q and raised to the power p or if you like a to the power p and then to the power one over q right it doesn't matter where you do whether you do the power first and then the root or the root and then the power right keeping in mind remember that a is positive here so we don't have to worry about well what if a is negative right that's not something that we're going to consider so we know how to deal with with rational numbers and now the question is well how do you how do you extend things how do you deal with a real number um and here's maybe where a little bit of calculus comes into the picture um and and so what we can say is well we'll we'll deal with real numbers maybe we could say by continuity and what do we mean by that well what we mean by that is that you can of course plot an exponential function right um so if we had so here's some axes and and so i'm going to plot let's say a is bigger than one so if a is bigger than one it's going to go up like this right so we know that a of zeros is one right and then you know we get that and then so so we plot our integer values right then you fill in the rational values and you get something like that um and and i mean it's actually better than this because you know the rationals have this property of being dense in the reels right in in calculus we usually don't get into these technical details but um one of the things that you can prove is that choose any two real numbers you want there's always a rational number in between them so the rational numbers are packed in there really tightly really close but we know that the rationals aren't everything they're irrational in fact they're in some sense more irrational numbers than rational um but you kind of the way you define it for for a real variable is by just kind of you know connecting the dots right so to get to get your exponential function of a real variable you fill in the gaps and the continuity just means that you know you you fill it in so you get this this continuous unbroken line right you're not going to you're not going to suddenly put a point up here you're going to follow the curve all right and that's that's one way of thinking about how you might define an exponential function of a real variable all right so in the last video we introduced exponential functions um so we said we can define f of x as a to the x where x is a real number we do have this condition here that a has to be positive right we we can't define exponential functions for a negative base because we know that we we run into problems with things like square roots of negative numbers right um infin in fact if a is negative then the only values that are going to work for for the exponent are going to be either integer exponents or rational exponents where where the denominator is odd right and things are in lowest terms so so you can't define exponential functions if the base is negative but for any positive value of the base you can make sense of this all right here are a few of the graphs right so for a bigger than one you get something which kind of grows like this right and then they grow very rapidly you probably hear this term exponential growth right referring to something that grows very very quickly it gets very big very fast and that's because these exponential functions do indeed grow very big very fast the so exponential functions grow faster than any of the other elementary functions that you're used to dealing with as the base gets bigger the growth rate is faster right at the at the positive x end at the negative x end they tail off towards zero so they get closer and closer and closer to zero but they never quite reach it right so so the x-axis is a horizontal asymptote but only on one side right so it's a horizontal asymptote as x goes towards minus infinity if you like okay if if a is between zero and one then you get this decreasing function rather than an increasing function right so it's big for negative values of x and then it it shrinks down towards zero of course you could also put a equal to one where that's not very interesting because one to any power is is just one um and so you in fact have a constant function so we're not so interested in that one of the ones that's that's not pictured which is the most common exponential function is the natural exponential and that kind of sits somewhere in here okay so here is y equals e to the x so this e is is euler's number um and it's it's around 2.7 okay it's is an irrational number in fact it's what's called a transcendental number just like pi so it's you know it's a in some sense a complicated number uh in some sense a simple number it turns out that for a lot of reasons the natural exponential especially from the point of view of calculus is is is the best exponential function to work with e is the simplest base despite the fact that you know it's an irrational number it's a simple base to work with we'll see that derivatives and integrals working with base e are much much much simpler than using base 2 or base 10 or one of the other log bases that you might have worked with in high school okay so properties what can we say about exponential functions well for any base the domain of my exponential function is is all real numbers so they're defined for every possible value of x right we saw that for for a bigger than one we can say a couple of things we can say that as x gets big f of x gets big two we can maybe we can say uh bigger all right it grows very rapidly and as oops as x gets big and negative f of x gets close to zero right so we have this idea of an asymptote once we have the language of limits we can we can state this much more quickly and much more precisely what we mean here um and for for a between 0 and 1 it's just the opposite right so for every for every number between 0 and 1 its reciprocal is is bigger than 1 and and if you take the reciprocal of the base you just reflect the graph across the y axis so you just get the behavior going in the opposite direction okay what else can we say about exponential functions well we have you know the algebraic properties that we've seen previously as properties of exponents so we know that f of x plus y is f of x times f of y right we know that f of x times y is the same thing as f of x to the power y and we know that f of minus x is 1 over f x which looks a little bit complicated when you put it in function notation but these are just the familiar rules right that a to the x plus y is a to the x times a to the y a to the x y is the same thing as a to the x to the y and a to the minus x is one over a to the x okay all right so those are the basic properties of exponential functions okay going into calculus the main the main things you need to be comfortable with for exponential functions is is this kind of basic knowledge of what the graph looks like right the fact that there is this asymptotic behavior at one end and rapid growth at the other end the fact that it's defined for all real numbers and these algebraic properties because certainly you'll you'll find yourself working with these algebraic properties at various times and you know it you can sometimes get yourself mixed up on these properties so some people will get mixed up and they'll they'll think that maybe this should be x times y um or you know or maybe they think that there should be a plus sign here it's easy to get mixed up on some of these rules if you find yourself getting mixed up you could always kind of come back and remind yourself well you know if if x and y are are natural numbers then this is just sort of repeated multiplication and you can make sense of things that way right if you if you find yourself stuck and you're not sure of the rules but that's that's mostly what you're you're going to need to be comfortable with is is these algebraic properties and and some good working knowledge of of the graph if you've got a handle on those you should be okay as far as exponential functions go okay next up we're going to talk about logarithms i've left the this graph up on the board here for exponential functions right so remember for for a base a where a is positive we can talk about these exponential functions right if a is bigger than 1 we have these ones growing like that right 2 to the x 3 to the x in between we have e to the x the natural exponential so logarithms the reason we're leading into logarithms from exponentials is that logarithms are are defined as the as the inverse of of the exponential right so so if i give you you know y is equal to a to the x and i want to take the logarithm base a of y what i get is x okay so in other words you have this cancellation property um which we know we we have in general for inverse functions right so we know that log base a for a to the x is equal to x and we know that if i did a to the power log a of x that also gives me x okay um so the logarithm is is defined as an inverse so another another way that you could say this is that um y equals a to the x if and only if x is equal to log base a of y or another way you might say this is if f of x is equal to a to the x then its inverse function is the logarithm so what a logarithm does is you feed it a number right so suppose you you feed it some number let's say this y that we have here right so you plug y into your logarithm so what does the logarithm do what the logarithm does is it gives you the answer to the following question that question being if i were to write y as as an exponent as a power if i would write y as a power with base a what exponent would i need right so what power do i have to raise a to to get y that's what the logarithm is answering for you right so that means that for example if i wanted to know what is the base 2 log of let's say 16 i say oh well how do i write 16 as a power of 2 and i say okay well 16 is 2 to the 4 and the logarithm gives me the exponent so my answer is 4. right so so that's the basic idea of a logarithm now most of the time you don't get lucky and your the number you're plugging in there is not a perfect power for the for the base that you're working with and so you're not going to get a nice round number you're probably going to get something fairly complicated right but it gives you a starting point it gives you some idea of what you're working with here the other thing you can do is you can now work out what the graph should look like for a logarithm right so if this is what the graph looks like for an exponential let's put it down here okay so if we have some exponential function like this so here is y equals a to the x okay here's the line y equals x which you might reflect across and so remember the basic idea with inverse is you're kind of you're interchanging the role of x and y here right um so if if if x y is a point on the graph of y equals a to the x then then y x is a point on the graph of the of the logarithm and and so everything kind of reflects across so that means that you know if we had so this point here is always 0 1 which means that 1 0 must be a point on the graph of the natural log okay as as x gets small as x goes to minus infinity y gets close to zero so we we interchange those roles as y goes to minus infinity x goes to zero so that means there's now this vertical asymptote along the x-axis so you have that sort of behavior going down and then it's going to come up and head off that way right so y equals log base a would look like that and of course if a is between 0 and 1 then then we we reflect everything across right that's going to go that way vlog's going to go that way um and we can deal with that as well so that's the basic definition of the of the of a logarithm log base a most of the time and let's just add this in as our last point here for notation and then in the next video we're going to look at properties so when you're working with the natural exponential the log base e in most calculus textbooks we write ln for natural log okay ln of x is the natural log so ln of x is the inverse of the natural exponential e to the x one one word of warning one word of caution um there there are some notational discrepancies in most calculus books that are sort of targeted towards science and engineering this is the notation you're going to see almost universally for the natural log in textbooks that are geared towards pure math students like math majors who aren't planning to do anything in the sciences sometimes if you see just log with no base indicated the assumption is that that's the natural log because for for mathematics for pure mathematics base e is really the only base that you care about working in you don't really care about other bases so you just stick to base e um and so you just write log for the log you kind of treat it as there's only one log but but in the sciences if you write log with no base sometimes that's interpreted as base 10. so if you see log without a base you really need to ask yourself what the context is am i in kind of a math context and then it's probably base e am i in a science context in which case it's probably base 10. right if you want to avoid the ambiguity and you're working with natural logs you can write ln to be safe okay so we're ready to look at some of the properties of logarithms now i'm going to focus on the natural logarithm here rather than general base a logarithm because once you understand the natural log it turns out you understand every every log every other log is is going to be basically the same as the natural log we'll we'll go into details on that explain why that's true towards the end of the video okay so basics the natural log is defined as the inverse of the natural exponential okay and so we can think a little bit about well what do we know about the natural exponential we know that the domain is all real numbers it's true for every exponential function we we saw from the graph that the range is from 0 to infinity right as x gets negative the graph approaches the x-axis but it never actually reaches it so the range is from 0 to infinity domain is r when you take the inverse domain and range they switch roles so that means that over here the the domain is going to be from 0 to infinity so in particular that means that the natural log is not defined for zero it's not defined for negative numbers is only defined for positive numbers right and the range well the range is all real numbers the range is from minus infinity to infinity right if you remember what the graph looks like this makes sense right let's just throw that in here okay so the graph for your natural log looks something like this right we have this intercept at 1 0. so all the all the negative values in the range are attained for x values between zero and one and the positive values are attained for x values bigger than one one uh one thing to point out is that this is a very slow growing function right it's the inverse of the exponential we said the exponential is very fast growing the logarithm is is is one of the slowest growing functions right i mean not not as slow as let's say a constant function or something like a sine function which is periodic it never gets bigger than one but it you know as x gets big the natural log will will eventually get big it will eventually go to infinity but it gets there very slowly okay and that's one of the reasons why people like using logarithms logarithms are useful when you're working with very large numbers they sort of tame those numbers down they give you smaller numbers that are easier to work with right um another reason that people like logarithms is they tend to take complicated arithmetic operations and turn them into simpler ones right so one of the properties that we have for the natural log is that the natural log of say a times b is equal to the natural log of a plus the natural log of b right so if you've got some collection of numbers that you're working with and you've taken the natural log of all those numbers then multiplication becomes addition right addition is simpler than multiplication it's easier to work with right why is that why is this rule true well if we use this association here right so we can see why this is true okay on on the one hand i know that the natural log of a times b would be the natural log of e to the x times e to the y if i set a equals e to the x and i said b equals e to the y but i have a property here for exponentials right we know that e to the x times e to the y is e to the x plus y okay so i can write it like that ah but i also know that if i take the log of the exponential of something those those cancel out right because they're they're inverses of each other right the fact that f and g are inverses right this means remember the definition of the inverse tells you that that f of g of x is equal to x for any input x so in particular for this input x plus y we have that all right so this is just x plus y all right um but what's x plus y well x is the natural log of a y is the natural log of b and that gives you the right hand side okay so similarly you could do this with division okay so the natural log of of a over b is the natural log of a minus the natural log of b and why is that well let's see the natural log of a over b would be the natural log of e to the x over e to the y which is well we know that e to the x divide by e to the y right another property of exponents says that's e to the x minus y so that's e to the x minus y and again log in the exponential cancel each other out because they're inverses i get x minus y so i get log of a minus log of b all right the last property which is which can be quite useful is that the natural log of a raised to a power let's say k is the same thing as k times the natural log of a okay so this can be quite useful right this tells you that that again right logarithms are simplifying your arithmetic it's taking a power right exponents is a fairly complicated arithmetic operation and it's just turning it into multiplication so things are simpler right and the reason why on this one well if i do the natural log of a to the k where a is e to the x again i have a property of exponents that says if i do e to the x and i raise it to some power that's the same thing as e to the k times x this is the natural log of e to the k times x and once again i use the cancellation property of inverses to get k times x and x is the natural log of a right so i get that property okay so those are the three main algebraic properties of logarithms right together with domain range this picture of the graph the other thing that's probably useful to remember is this intercept log of 1 equals 0. this is actually true for any log for any log if you plug in one you get zero right any base that's the result okay now i mentioned that this is really all you need to know um is is these properties and you need to know them for the natural log the reason that's all you need to know is there's a formula that lets you work with logs in other bases there's this so-called change of base formula i think we'll look at that in the next video okay so in the last video we introduced the natural logarithm we went over some of the properties we showed how these properties followed from properties of exponents right from the exponential function in its properties um and and i mentioned that you know as long as you have these properties and as long as you kind of know what the graph looks like you you know pretty much everything you need to know to work with logarithms right again it's easy to make mistakes with these properties there's always some wishful thinking that there should be a rule when you have addition inside the logarithm unfortunately there isn't so some people will get these mixed up it's okay it happens you get the hang of things with practice but the one thing that does come up is okay what if you have what if you have some other base right i did everything for the natural log so what if you're working with say base a right so let's say what about let's say we're doing f of x equals log base a of x okay we want to understand how to work with this so it turns out once you understand the natural log you pretty much know everything because every other logarithm can be written in terms of the natural log to see how this works let's let y equal to base a log of x okay now remember what this means saying that y equals the base a log of x is the same thing as saying a to the y is equal to x right okay now here's a here's a trick that you can do remember that remember that the natural log is the inverse of the exponential function right and that means that um oh maybe not f since e to the x and ln of x are inverses that means that if i did e to the log of x they cancel each other out they give me back x so in particular i could do a to the y would be the same thing as e to the natural log of a to the y okay and that in turn was equal to x okay now remember that we have this property down here right we can bring the k out front okay so i can bring that y out front so i can say that e to the y times the natural log of a is equal to x okay now let's take the log of both sides and see what happens the natural log of e to the y ln a is equal to the natural log of x right natural logs of functions so equal inputs have to produce equal outputs very good um but since these are inverses it's also true that if i do the natural log of e to some power again those cancel out and just give me back the power so that means that what i get is that y times the natural log of a is equal to the natural log of x and remember what y is y was our original base a log so i can solve for y here and what do i get i get that y which is the base a log of x right and from there i can see that y is the same thing as the natural log of x divided by the natural log of a so i get this okay this is the so-called change-of-base formula okay and what it tells you is that a logarithm and any other base can be written in terms of a natural log so if you understand the natural log you understand every other logarithm and again from the point of view of calculus we're going to see that once we get into derivatives integrals things like that even limits the natural log is is easier to deal with than logs to other bases and so we try to do everything in terms of the natural log all right we're going to finish our review of logarithms with a couple of quick examples first we're going to look at some of the graphical properties and then we're going to play around with some of the algebraic properties okay so here's a function f x is the natural log of 3x minus 6. we want to figure out the domain so we come over here and we notice that for for the sort of basic natural log with just x as input the domain is x has to be between 0 and infinity so that means the input has to be bigger than 0 right so when we come over to something like this it's no longer the case that we just need x to be bigger than 0. we need three x minus six the whole input has to be bigger than zero right that means that we need three x to be bigger than 6 so that means we need x to be bigger than 2 right so we can leave our answer like that if you like x bigger than 2 that's your domain if you prefer you can write that as the integral from from 2 to infinity so that means that if you were going to try to plot the function okay well the first thing you're gonna need is you're gonna need a vertical asymptote now at two right so there's a vertical asymptote where the domain begins okay all right so essentially we've we've shifted it over right um one way to think about it is this is 3 if you factor out the 3 right 3 times x minus 2. so if you think about it that way what we've done is we've shifted to the right by two units thinking through the transformations we've shifted to the right by two units and then we've uh we've compressed horizontally by a factor of three all right um and so all that's gonna do is that's gonna is gonna kind of rather than having it go like that it's gonna be a little bit steeper right the general shape is still the same okay uh we can still work out if we want to know what the intercept is we can say well we know that the intercept is going to happen when the input is equal to 1 right we know that we know that the natural log of 1 is always equal to 0. so if we set 3x minus 6 equal to 1 so 3x equal to 7. so at x equal to 7 over 3 right which is what just two and a third so it's it's around there all right so you can see it's it's compressed by a factor of three right rather than that intercept being one unit over from the asymptote it's only one a distance of one third from the asymptote and then we can we can plot it in so we're gonna have something that looks like that okay there's our graph notice that it's the same general shape right it's just a transformation of the original okay now here's a here's an algebraic problem this is working with the properties we're given three values for the natural log log of a is two log of b is minus one log of c is three we wanna find the value of this ugly looking expression here okay now one of the mistakes that people will make this becomes a complicated question if you if you think that well if you know if you think that this is telling you that your first step should be to solve for a b and c you're going to have a terrible time with this problem and yes i could solve for them right a is going to be e to the 2 b is going to be e to the minus 1. i could i could do that but but the point is that i don't have to do that because i have these properties the point is these properties let me break things down one thing that you you want to watch out for i put square brackets around around the one in the denominator to make it clear that that power applies to the logarithm and not to the input this power rule here log of a to the k this applies when the exponent is inside the logarithm not outside one of the the problems is that we sometimes encounter ambiguous expressions like this now in this case the intent is that that power is inside the log usually that's sort of the assumption if it's if it's not clear if you're ever unsure you could always ask right be like hey this is ambiguous can you clarify okay so with something like that we can on the top we can use the exponent rule right and we can write this we can write the top as 3 times the log of a cubed b to the 5 over over c to the 4. the other thing we could have done is we could have applied the x we could have used laws of exponents on the inside right a to the 9 b to the 15 c to the 4. it's going to work out the same in the end whichever way we do it but that's fine and on the bottom there's actually not much you can do right you just leave it as is one of the things to be careful about here is there's going to be a temptation to bring that power down in front but because that power is outside the natural log it's outside the function there's nothing you can do about it other than work out the value on the inside and then square okay now what do you do from here well with something like i'm going to work over here because we don't have that much room with something like log a cubed b to the 5 over c to the 4. we could apply the the quotient rule and say well that's like the natural log of a cubed b to the 5. subtract the natural log of c to the 4 right and then you could apply the sum rule to that first term right and you could write it as log of a cubed plus the log of b to the 5 minus the log of c to the 4. once you get the hang of this you'll realize that it's always going to be the case that that terms in the numerator just come with plus signs terms in the denominator denominator come with minus sign right when you have things factored like this and then finally you can bring the powers down right so what you're going to have is you said that 3 out front you're going to have 3 log a plus 5 log b minus 4 log c okay on the bottom you can use that same reasoning to say well i've got 2 log a plus log b minus 2 log c and that whole thing is still squared okay and from here you can put your numbers in um i'm running out of boards so maybe i well i can put it just above so what do we get we get 3 times 3 times log a so 3 times 2 6 5 times log b so 5 times -1 6 minus 5 minus 4 times 3 minus 12 and on the bottom 2 log a so two times two four okay minus one minus two times three so four minus one minus six and i want that squared okay all right so minus 33 over what's that going to be on the bottom uh over 9 which i suppose you can simplify to minus 11 over 3. right but the point is to use the properties of logarithms break everything down then put in your numbers right it's primarily an exercise in making sure you know these properties making sure so the other pitfall here when you have a log divided by a log there's going to be a temptation to try and apply this rule and subtract the two but that only works if the division is inside the logarithm not outside right all these rules apply when the operation right the multiplication division or the power is inside the log and then you can split it up into something similar simpler on the other side all right so we've been doing a lot of algebra we're going to take a short break from that we're going to talk about graphs for polynomial functions right so we're looking at polynomials we've been looking at things like how to factor them right things like long division all this stuff algebra manipulating polynomials what about graphing right later on when we're doing calculus there's going to be a fair amount of graphing involved what do polynomial graphs typically look like well it helps to first think about what do power functions look like okay so what if i have a function that looks like f of x is equal to a times x to the n okay so n here could be 1 2 3 and so on okay well depends somewhat on whether a is positive or negative okay but let's do n equals 1 two three four we'll do a few just to just to get a feel for for what's happening okay so coordinate axes in each case there we go all right now when n equals 1 of course in that case you're just dealing with a linear function right x to the 1 we get a straight line right a straight line with slope a intercept 0. so it passes through the origin right if a is positive positive slope so we have something [Music] like that let's say if a is negative negative slope we're dealing with something like that n equals 2 that's your basic quadratic opening upwards if a is bigger than zero opening downwards if a is less than zero okay n equals three your basic cubic looks like this it starts negative it's going to flatten out as it passes through the origin and then head up so it's going to look like this okay and if a is negative same thing but flipped okay cubics are going to look like that degree 4 degree 4 looks a lot like degree 2. it's just going to be a little bit steeper on the edges a little bit flatter on the bottom we get something that kind of looks like this okay something like that for degree four okay um degree five you're going to look a lot like degree three except again steeper out here flatter in there degree six gonna be like degree four but it's gonna kind of be again a little bit steeper on the sides a little bit flatter on the bottom and so on right so in general all the even degree ones are gonna look something like quadratic all the odd degree ones are going to look something like cubic okay now it's important to know what these power functions look like because in general right you're going to be looking at a function of the form say a n x to the n right give your leading term plus maybe some lower degree stuff right general polynomial function now that leading term that's the dominant term right so this this is going to be sort of the most important term when the absolute value of x is big okay so are there large and positive or large n negative right so as you kind of head out right your polynomial is going to look a lot like one of these so the so-called end behavior if you like what happens eventually is determined by that leading term okay everything else controls what's going on in the middle right so if you've got additional terms other than that leading term that's going to sort of you know spice up things here in the middle right you might have some roots right so we know what it kind of looks like when x is big and when x is small so if we're kind of nearer to the origin that's where we want to look at the roots here so as an example let's say we have a polynomial and i'm going to factor it for you let's see how somebody looks like x minus 1. times x plus 2 squared okay that's our polynomial what is that going to look like well let's draw some axes so first of all we know that if we were to multiply this all out and we don't have to multiply everything out but we can see without doing all the work that the leading term right there's going to be x squared here times x so the leading term is going to be x cubed right plus some other stuff so we know that eventually it's got to be doing got to be doing something like that right it's got to look like n equals 3 because it's cubic right but you might have some other stuff going on in between so that's where you look at the zeros so this thing touches the x-axis twice it touches it once when x is equal to one so let's mark off say one two three so there's a zero there minus one minus two minus three the other zero is that minus two so what i know is that i'm going to come down i'm going to hit that zero so what happens at that zero well one of the things that i can do is i can kind of on the side i can do a little number line i can mark off those two zeros all right minus two and one those are the two roots for my polynomial okay um i know that it's going to be positive out here i know it's going to be negative out here okay all right because the leading term is x cubed i know it's going to look like that these are the only places where the sign can possibly change right these are the only places where we might be crossing the x-axis the only places where we might be changing from positive to negative so this is either negative or positive in between how do we decide well one way we could do it is we just plug in something in between like zero and say what do i get well i get minus one times four that's negative okay so i know it's negative in between the other way to see it is to realize that because this term is squared right this will never be negative you won't get a sign change at minus 2 because this can't be a negative this term which has an odd power that's going to change sign the zero right so we know we get a sign change at one but not at minus two so we get this sort of sign diagram so what this tells me is that i should be crossing at one and now i maybe get rid of this because it might be hard to connect things up so i want something that is going to cross it's going to cross that one the other thing and by the way the other thing i can get is i can get that y-intercept right these are my two x-intercepts what's the y-intercept we work that out when x is 0 it's at minus four so it's down here right so i know that it's going to go up like that and i know it's got to come back up there the one thing i don't know is i don't know exactly when that's going to turn around that's where calculus would come into the picture calculus will let us find the exact location of where this thing bottoms out it's got to hit a bottom and it's got to come back up right it's got to come back up because it's got to hit that root at 2. okay but i don't cross at 2 right because this thing doesn't change sign right i have to be negative on both sides so i just come and i kiss the axis and i come back down right and i've got most of my graph the one detail i'm missing is here right so i need some i need a bit of calculus to uh to figure out exactly what's going on there but you can apply these basic principles even once you're doing these like you're you're doing curved sketching you're in your calculus course you're trying to figure out what the graphs and polynomial looks like you're lost in derivatives and second derivatives and and intervals of increase and decrease in concavity and all of this remember that you can get most of your polynomial graph just by finding the roots if you can looking at the leading term to get the end behavior and all that calculus is going to do for you is fill in a few details in the middle that you can't quite get just by looking at the zeros and the leading term okay so we've we've got our rational expression here we talked about how to factor simplify determine domain if we think of it as a function there are two different things that can go wrong as far as points that are left out of the domain in a rational expression or function like this okay this 0 here right there was a 0 in the denominator but i canceled it with a 0 in the numerator and now it's gone there's no longer right so once i simplify there's no longer a zero in the denominator when x equals zero these sort of zeros in the denominator that you can cancel these produce just simply a hole in the graph if you think about the graph of your function okay um so you know we had this 0 over 0 thing but once you cancel it right now i could plug in x equals 0 and in fact i get well it almost looks like an intercept except it's not quite an intercept because it's a whole okay what about at 2 well any zero in the denominator that does not cancel with a zero in the numerator this produces what's called a vertical asymptote okay so that's information that you can get out of the factored form for your rational expression as long as you keep track of the fact that oh there was that x that i cancelled right we took we kept track of that long as you made that note you have this information um the other thing that we know is that there is an x intercept right when x is equal to minus one y is equal to zero right so if we're again if we're thinking about graphing we know that our graph crosses the y-axis or crosses the x-axis rather at minus 1 okay at 0 so well at 0 0 we would have an intercept there it would be both the x and the y-intercept except our original expression was not defined at zero right so instead of getting an intercept there that's where we have this hole okay so we can get that information there's one more piece of information that you can look for in a rational function which is there's a second type of asymptote right rational functions they have both vertical and sometimes horizontal asymptotes right they don't always have vertical asymptotes because i could have like an irreducible quadratic in the in the denominator let's say right something that has no zeros that could happen this one does not have a horizontal asymptote we'll do another example where where there is one and we'll talk about how the way you figure that out is you you look up here um so what you can do is you can also again you can ask about what what's the end behavior what happens when the absolute value of x is big so when the absolute value of x is big our function behaves roughly like well what happens is when x when the absolute value of x is big enough you can ignore the lower order terms they're not so important you keep the top degree terms on the top and the bottom and so you have x cubed over 4x squared and again we're thinking big absolute value of x here so we're away from zero so we don't worry about the fact that this is undefined at zero if we simplify this is just x over 4 right so that tells us that the graph of this thing once x gets big enough it's going to look more or less like just simply the line y equals x over 4. okay in general when you want to know what's happening you know for large values of x this is what you do right you just look at the top powers in this case the degree of the polynomial in the numerator was greater than the degree of the polynomial of the denominator if the degree in the numerator is less than or equal to the one in the denominator you're going to have a horizontal asymptote so it doesn't happen here but it will happen in other situations okay so all of this is information that you can extract um just by sort of looking at the function right again we haven't done any calculus anything like that all we did was well we did one estimate by thinking about what happens when x is really big and we did a bit of factoring okay so we're going to take that information in the next video and we're going to see how to put all this together and get some rough idea of what the graph of this thing might look like okay so here's another example with a rational function where we're going to try to get out all the information that we can and see if we can draw a graph so one of the things we might do before we even try to factor is say well when the absolute value of x is really big this thing is approximately x squared over 2x squared which is just one-half right so that value there that one-half this is a horizontal asymptote okay so y equals one-half that's a horizontal asymptote for vertical asymptotes we need to factor so we say okay top is the difference of squares x minus one x plus one bottom x is a common factor actually 2x is a common factor so take out the 2x we're left with x minus 2. okay so right away from here we can see that x cannot equal to zero and it cannot equal to two um so that means that uh x equals two and x equals zero these are vertical asymptotes okay right neither of these zeros in the denominator cancels with something in the numerator okay so we know that those have to be vertical asymptotes we also know that that 1 0 and minus 1 0 these are x intercepts right those are places where the graph is going to cross the x axis because if x is plus or minus 1 y is going to be 0. okay there is no y-intercept because we can't put x equal to 0. there's a vertical asymptote there right so the y-axis is actually an asymptote in this case okay so this is pretty much all the information that you can extract without doing calculus or anything like that so at this point let's see what the graph looks like let's draw let's draw some axes let's draw some asymptotes so we have so here is say one all right so at one half we've got that horizontal asymptote okay the y-axis doubles as a vertical asymptote so does x equals 2. so x equals 2 is a vertical asymptote i know i have intercepts at 1. and at minus 1. okay so i have all that information so the other thing i probably want to do is i probably want to look at the sign diagram okay so we'll draw our number line we're going to mark off intercepts at minus one that plus one asymptotes at zero and at two um once you once you've done enough of these you start getting the hang of the fact that if if none of the factors are even powers then you're going to get a sign change at every single one of these points that you've marked off so you really only have to test kind of out past two try say x equals three you see okay everything is positive so we expect that we're going to get something that looks like that okay so this is useful because it tells me what's going on at the two vertical asymptotes it tells me that i've got a head down to minus infinity on the left of zero up to plus infinity on the right and same sort of thing at two okay i know i've got that going on the the one thing that i don't quite get from the sign diagram and i may not be able to determine exactly is am i going to approach this horizontal asymptote from above or from below on either sides that's that's a little bit trickier to work out but we have this bit of information here so what we know is that we have to kind of head down to that vertical asymptote there we have to pass through one we have to head up there so it seems like the most likely scenario [Music] is going to be something like that right okay probably looks like that in between there there's some possibility well not really i think you know that it could kind of go down and then back up but i mean we need to look at the the derivative to rule that out but chances are it looks something like that okay now here i'm just coming down now i know i don't cross the x axis again because there is another intercept so most likely i'm going to come down and just approach that horizontal asymptote again there's some small possibility that i dip down just below the horizontal asymptote and come back up calculus would tell me whether or not that would happen because if that happened there would have to be a minimum value i would find that using calculus notice by the way you are allowed to cross horizontal asymptotes vertical asymptotes you can't cross because those x values are not in your domain there's nothing stopping you from going across a horizontal asymptote okay so the last one is minus one so now the the only thing here is i know i've got to come up i've got to pass through that intercept and now the question is do i go up come down and go like that or do i come up and just kind of go like that all right i don't know which of those two possibilities it is again that's where calculus would come into the picture taking the derivative would tell me that which one of those two is it gonna be i don't know yet right we won't know until we we move on we do a bit of calculus we learn how to deal with that but again a lot of the shape you can you can work out right you can get some idea of the shape just by by looking at the function by factoring looking at this sine diagram looking for asymptotes intercepts you can get a pretty rough idea of the graph in fact if i if i really wanted to figure out what was happening here i could probably just try you know a couple of large x values and see is the y value bigger or smaller than one half when x is really big right that's one way that i could do it if i find that things are a little bit less than one half chances are it's the yellow curve if i find that things are a little bit bigger than one half chances are it's the pink curve right so i could i could figure that out if i had to okay so we introduced the notion of a graph in the previous video i left the definition up here on the on the board right so somebody hands you a function right you can look at all the ordered pairs a b where a belongs to a b belongs to b right all the ordered pairs where b is associated with a through the function right and and sometimes this graph is not something that you can necessarily plot right if a and b are not sets of numbers you can still talk about the graph even though you can't necessarily draw it if a and b are sets of numbers then you can then you can visualize the graph using using this idea of the cartesian plane right so for this example here right here here's an example where i would just plot the points and that's going to tell you what the graph is right so if i want f of 1 to be 3 so i go x is 1 right this is my x-axis this is my y-axis or a and b if you like and i plot a point at the coordinates one three f of two equals two so i go to two and i plot a point there okay f of three equals three so i go back up to three for my y value and i plot a point and f of four is equal to minus one i plot a point down there and i have my graph right it's not very exciting but but that's the graph um and this is indeed the graph of a function right again the way the way i would know that this was not a function is if there was more than one y value for one of these x values right so if i plotted another point say here at maybe one one or at one minus one down here or a one two i would no longer have a function right again it's okay to have two x values that both go to the same y value i just can't have two y values associated to a single x value okay so that's a function with that in mind we can come to these sorts of plots that are more like what you might see in a calculus course and you can ask okay which of these are graphs and so one of the things you probably learned in high school is that the thing that distinguishes a function from all other similar types of objects all of the curves if you like if you think in terms of the graphs is that the graph of a function passes this so-called vertical straight line test right so the idea is that if you draw a vertical line it should only cut the graph once right so no matter where on the graph you go you draw a vertical line and it only cuts once there's this notation convention we use for graphs if i fill in a point it's included on the graph if i leave it hollow it's not included so here i draw that vertical line down all right passing through those two points and that still only cuts the graph once because that point is not included right so so this one would be a graph this one here clearly is not because if i draw a vertical line here i see that i cut it in two places so it fails that test right even though even though this graph comes in two pieces that are not connected right um standard example that people usually give for for a curve that is not a graph of a function is a circle right circle fails this vertical line test but in fact if if if i had even one sort of extraneous point if i plotted a point here and said oh that's part of the that's part of the graph as well this point uh and the curve i'm not dealing with a function right something like this so this one is not this one down here also not a function at least if if all of this is supposed to is supposed to be two parts of a single graph it's not a function right because even though this part here is looks like a line lines are fine lines are graphs of functions this bit here again it fails the test so we don't have a function right maybe i could have modified this one so that i only include say that the top half of this parabola not the bottom half then maybe this is a little bit ambiguous because you'd then have to start wondering like oh does this line keep going because if it keeps going eventually it's going to overlap that and again i wouldn't have a function so sometimes you know the the picture that you draw the graph might be an incomplete picture right maybe it's not telling you everything you need to know about the function you might have to go back to the formula right to know for sure whether you're dealing with a function but the way to know whether or not you're dealing with a function is really just you know you look at the formula you look at the graph you look at whatever information you have that's defining your function and you ask yourself is there any input is there any element of the domain for which i can get more than one output if there is you don't have a function okay so in this video we're going to look at some common graphs that we might encounter okay um just to give us an idea all right and basically the principle is you start with the simple examples and you learn how to build up from them to more complicated ones so common graphs well we have lines right so one of the ways you often see lines written y equals mx plus b this slope intercept form right so the graph of the line tends to look something like this right this point here the b right is that y-intercept right the place where it crosses the y-axis m the slope rise over run and again with with with graphs of lines right visually you can you can pretty easily tell a line with positive slope apart from a line with negative slope or or a line with zero slope which is just a horizontal line right but but from the graph you probably can't look at it and read off the exact value for the slope for that you're going to have to either get a couple of points calculate rise over run or maybe you have the formula handy and you can look at that right so we have lines we have basic quadratic right so this is a parabola opening upwards vertex at zero zero something like that okay we could go to the cubic the graph of the cubic looks something like this starts down here comes up flattens out as it goes through the origin and then it heads up okay with the cubic and then other power functions integer power functions tend to look like variations on these you could also look at say root functions so we could look at say y equals the square root of x now of course here there's a domain issue right this is only defined when x is bigger than or equal to zero so we can't we can't plot it for negative x for for x bigger than or equal to 0 it looks like this in fact that root function is is just one half of a parabola but turned on its side right and again this this is related to this inverse relationship right uh the square root is is sort of a partial inverse for the for the squaring function and this is one of these places where domain does come in into play right this function here if you set the domain to be all real numbers does not have an inverse but if you were to restrict this to only x bigger than or equal to 0 for inputs so you only took this half of the parabola then this graph and that graph they would be inverses of each other we probably won't deal with inverse functions in the review i think this is something we'll probably leave until we get through a bit of calculus and we want to talk about derivatives for inverse functions we'll we'll deal with inverses when we come to them okay so we have some of these basic algebraic functions um i guess maybe one more we could put in here before we move on might be this basic hyperbola y equals one over x which looks something like this has two pieces and it has both horizontal and vertical asymptotes um this is sort of a it's both a good and a bad example um it's good in that it has an interesting behavior that you don't encounter with things like power functions the bad thing about this this example is that some people will if you spend too much time on this some people will kind of get the impression that whenever you have asymptotes they're always the axes this is not always the case right a horizontal asymptote could be any horizontal line a vertical asymptote could be any vertical line when we get to graphs of rational functions that's something that we're going to encounter okay other functions that you will encounter in this course there are the trig functions maybe i shouldn't try to do all the trig functions let's do sine so sine is this function which oscillates between -1 and 1. so this is an interesting property that we don't see with any of the ones over here right with all of these graphs the y value tends to either increase or decrease with the x value and and it tends to be that the y value will get arbitrarily large if you go out far enough one way or the other for a sine function the y value is always between minus one and one so you get a curve just goes back and forth forever and it repeats itself what's also interesting and this is true of all the trigonometric functions the sine function is what's called periodic the graph repeats once you know the graph from say yeah we could go from minus well let's say from from minus pi all the way to pi this is minus pi over 2 that's pi over 2. once you know that bit of the graph for the sine function you can just copy paste to get the graph of the sine function for all other values of x which again is not something that you see with any of these over here and so we can get into cosine tangent cotangent we could get into all the trig functions but we'll we'll probably deal with that when we when we go over trigonometry another one the exponential function i'm going to do e to the x but you can do other bases as well okay so the graph of y equals e to the x [Music] looks something like that okay the intercept is 0 1 right because anything to the power is 0 equal to 1. so pretty much i've done this for base e but for any base bigger than bigger than 1 this is what your exponential graph is going to look like if the base is between 0 and 1 it's going to go the other way okay but it's something which grows very rapidly towards infinity as x gets big and positive and it slowly decreases towards zero as you as you feed in negative values for x and the last sort of common function whose graph you should know is the natural log and the natural log is the inverse of the exponential function and again if you're doing a log to another base the graph is going to look the same just kind of stretched a little bit stretched or shrunk so because of the inverse relationship between these two functions the y-intercept for the exponential function we flip the coordinates and we get an x-intercept for the natural log this horizontal asymptote becomes a vertical asymptote and we get something which looks like that okay and again the natural log has this domain issue it's only defined for positive numbers it's undefined if x is 0 or negative so we only get a piece of the graph that looks like that if you wanted to if you wanted to do the absolute value of x actually we should throw the absolute value function in there if you do the natural log of the absolute value of x then you get sort of a a mirror image on that side right you get that piece and and you get that piece okay um if we want to just plain old y equals absolute value x let's squeeze that in right here in the middle uh the absolute value function has this sort of v shape right when x is bigger than zero it's just the line y equals x so it's just a straight line going up like that when x is less than zero y equals minus x so it's a line with with negative slope you get this v shape the absolute value is interesting it's an interesting function it's kind of the simplest example that most people come up with for a function which is continuous at every point once we define what it means for a function to be continuous but does have a point where it does not have a derivative right at the origin there's no well-defined slope for that function because the slope abruptly changes from minus one to plus one okay so these are some of your your common graphs that you're going to encounter throughout the course in the next video we'll talk a little bit about how to take some of these basic graphs and turn them into things that are slightly more complicated all right so in the last video we looked at some of the basic graphs i left a couple of them up here on the board right a basic parabola this one is also a parabola i'm coming from the root function it just happens to be a half of a parabola opening horizontally right rather than vertically there are a number of basic transformations that you can apply to graphs and and you can of course you can consider these in combination so you can start with a simple graph and you can apply some number of these in sequence to produce a new graph right so one of the things that you will notice if you kind of compare what's going on right i've given sort of the what happens to the function in each case you'll notice that for horizontal effects you're applying something inside the function right you're you're subtracting a number from x you're multiplying x by a number you're putting a minus sign in front of x for vertical transformations it happens outside the function right you do f of x and then you add b you do f of x and then you multiply you do f of x and then you apply the minus sign okay so horizontal is inside vertical tends to be outside so the the translations if you're if a is a positive number something like x minus three for example that's a shift to the right if a is negative you're shifting to the left if it's a stretch you want to look at so let's let's focus on a bigger than 0. you want to compare if a is between 0 and 1 or is it bigger than 1 okay if a is between zero and one then what tends to happen is you stretch it out like this if it's bigger than one you kind of squeeze it in okay um reflection if you put the minus sign in there you're reflecting across the y-axis okay and then same idea for vertical translation you're shifting up or down for the stretch if b is bigger than one you're making it bigger if it's smaller than one you're squishing it down and reflection you're reflecting across the x-axis so for example if i did something like y equals x minus a well let's put a value in x minus 9 squared and i plotted that what i would do is i would take my usual parabola and i would go nine units out and i would draw the graph there i'm going off the screen but that's okay you get the idea so you just take the usual graph and you'd slide it over right if i did something like y equals now here's one where maybe you've got to be a little bit careful with the stretches because let's say we do something like 2x squared um so this is one of these kind of odd situations where where a horizontal stretch kind of becomes a vertical stretch because you square the 2 you get 4x squared right so it'd be similar to if you just took the x squared multiplied by 4. but what tends to happen in this case is you kind of get the same graph but now it's kind of narrower right that horizontal stretch it kind of took this and it squished it in a little bit to get a narrower version of the original graph now if i do f of minus x for for x squared nothing happens right because you square you square minus x you get x squared this happens to be an example what's called an even function f of minus x is the same as f of x so you don't see any effect in this case from a horizontal reflection right reflecting that across the y-axis nothing happens but i could do if i did if i use the root function instead if i did say y equals the square root of of negative x well now i can't use any positive inputs because the minus sign out front makes it negative i have to use negative inputs and what i get in this case is something that looks like that right it goes the other way now instead if i wanted to in this example here maybe we'd do another example uh if i did something like y is the square root of x plus 1 well that's going to take my usual square root function shift it one unit to the left and give me something going up like that right and for the vertical translations i could do something like you know i could do y is equal to say x squared minus 1 or i could do y is equal to let's say root of x plus 2. and if i if i were to plot those this one i'm taking the regular parabola and i'm shifting it down by one unit so i get something like that i'm taking the square root and i'm shifting it up by two units so i get something like that right starting at two so with some basic with some basic translations stretches reflections you can start with some basic graphs you can turn them into new graphs and of course you could do combinations of these so i could do something like this i could do something like y equals 2 times x plus 1 squared minus 3 right so this is now several transformations all in one in fact let's even do minus 2. so what would i get if i if i had that well i would start with my basic parabola my basic squaring function the plus one shifts me one unit to the left the two is going to stretch it by a factor of two the minus sign is going to flip it over and then that minus 3 is going to shift everything down by 3 units so if i were to plot that let's say over here so now my vertex moves over right it's going to move over 1 unit right and then it's going to move down by three units and i would get something like that if i were to plot that function right so if you if you understand the effects of these basic transformations and you know some basic graphs then somebody can hand you a complicated function like this or looks a little bit more complicated um and you still know how to plot it without having to do any calculus or or really all that much work other than knowing some basic examples and knowing what happens when you apply these transformations what the effect of those transformations are on the graph okay so we're going to look briefly at the graphs for the six trig functions um primarily sine and cosine we'll try to get tan done maybe cotan we'll see how we do with secant and cosecant to be honest if you don't remember what those graphs look like it's probably not going to affect you in any way but let's start with let's start with sine so we know that sine has zeros add all the multiples of pi right we know that it hits its maximum value at pi over 2 right sine of pi over 2 is 1 all right at 3 pi over 2 it's down at minus 1. same thing at minus pi over 2 minus 3 pi over 2 it's back up at 1. okay and the sign the sign graph is all often referred to as a sine wave it sort of gently oscillates back and forth between these values so you get something that looks sort of like this okay okay and then that graph just keeps repeating forever so it just keeps oscillating up and down okay so this is the graph y equals sine x okay if we're taking this as our x-axis and this as our y-axis all right on the same set of axes we can plot cos right so cos has its zeros at the odd multiples of pi pi over two rather cos of one or cos of zero rather is equal to one cosine of pi is negative one at two pi where we're back up to plus one at minus pi coses at negative one and at 2 pi we're back up to plus 1. okay and the cosine graph goes in much the same as the sine graph okay like that okay so that's y equal to cos x okay in fact one of the things you might notice is that the graph for cosine is just a translation of the graph for sine right the sine graph is just shifted over by pi over 2 and that's that's no coincidence right so so one of the things that you'll notice is that um sine of of x is the same thing as cos of x minus pi over 2. right um so that's that's one of the things that you might notice right so when when when x is equal to zero right um cosine is equal to zero when when x is equal to pi over two we're at one yeah so that works out okay so you do have this this kind of translation property between the two of them we have this relationship all right now let's look at tan theta so um tan theta we notice that there are these gaps in the domain right and i mentioned that those gaps are vertical asymptotes so for tan first thing we have to do if we're going to plot 10 is we gotta mark off those vertical asymptotes okay okay so there are those asymptotes for 10. zeros are the integer multiples of pi okay uh and the other thing you got to work out is is which way is 10 going on either side of those asymptotes um and so we know that it's positive in the first quadrant you can work out that it's it's negative here in the fourth quadrant and in fact what you're going to get is is that it looks something like this so from minus pi over 2 to pi over 2 you get something that looks like that and and then this graph just repeats okay okay so that's what the graph looks like for the tan function okay uh for cotangent you're to get a graph that's similar but because you're flipping things over the asymptotes in the zeros those are going to trade rolls right so now you're going to have asymptotes at multiples of pi you're going to have zeros in between and i think maybe just to to keep this video from getting overly long we'll we'll skip graphing kotan if you're curious you can always pull up you know geogebra or desmos or any online graphing calculator and just fire it in and see what it looks like and you'll see that it is again very similar to the graph for 10. now for um for secant and for cosecant let's take a look at secant let's say and then again for cosecant that because of this relationship between sine and cosine there's a corresponding relationship between secant and cosecant so if i'm doing secant let's say so one of the things i want to do if i'm doing secant is i want to notice first of all there are these they're going to be these asymptotes again because i'm dividing by cos okay right there are no zeros there are no zeros secant is always bigger than or equal to one in absolute value okay so what you're going to get is you get a point here okay so when cosine is equal to one secant is equal to one one over a one and then as we approach those asymptotes cos gets closer to zero and so it's gonna head up to infinity because you're dividing by zero okay then over at at three pi over two you're at minus one and it heads down then at two pi you're going up okay and same thing here down then up okay so you get a graph that looks something like that so this would be y equals secant x um because i forgot to label this one this is y equals 10x okay another thing you might notice worth pointing out cosine and secant are even functions right the graph to the left of the x-axis is just the mirror image of the graph on the right right you can reflect across if you do this reflection like that nothing happens the other four are all odd functions so if you reflect so if you take say the graph of sine if you reflect across and then you flip so if you're if you're reflected across both axes um you will get the same thing right so you'll see that things are kind of opposite on either side right here they're they're the same here ones above one's below but you have these mirroring properties um for these so those will come in handy those properties might come in handy for various problems that you're trying to solve it's quite frequently useful to be able to remember that cos of minus x is the same thing as cos x and that sine of minus x is in fact minus sine x and these if you want you can take these as as identities and they're the first of several identities that you might need to use in your calculus course and we'll look at a few more in the next video okay so we're going to start a review of trigonometry with right triangle trinketometry so this is usually where people start it's sort of the simpler way of looking at things looking at right angled triangles and so you probably have seen definitions of trigonometric functions as ratios of the size of this triangle right so typically what you see is you see things like for that given angle there that interior angle you see things like sine of theta and sine of theta is given as the ratio of the side opposite the angle over the long side which is called the hypotenuse right so this c is your hypotenuse okay cosine is the adjacent side a or with the hypotenuse so it's the ratio a over c and there's also tangent tan theta and 10 theta is the ratio of the opposite side over the adjacent side and one of the things that you'll notice is that's the same as doing sine theta divided by cos theta okay those are the three basic trigonometric functions okay but there's uh there's one other character that usually comes into the play and that's pythagoras so we have the pythagorean theorem and the pythagorean theorem is the statement that a squared plus b squared has to equal c squared okay that's the pythagorean theorem and there are lots of so-called pythagorean triples examples of of integer values for a b and c that fit this equation the most commonly known one is is your three four five triangle right so three squared plus four squared uh is nine plus sixteen which gives you 25 which is five squared um some of you may have heard of fermaslav's theorem which says that if you go for any higher integer powers here cubes are greater it's impossible to find integer solutions to this equation so something special about the second power pythagorean theorem there are lots of examples of these triples of integers that fit the equation which is interesting but of course most of the time if you if you choose two integer values let's say you choose integer values for a and b chances are c is is not going to be an integer it's probably not even going to be rational it's going to be some square root um this this was apparently something that was a little bit troubling for for the greeks who really wanted to believe that everything could be expressed in terms of integers and ratios of integers one way to think about the pythagorean theorem you can think of it as as a relationship between areas between the area of you know a square of side length b a square a side length a if you add up the area of those two squares you should get the area of a square with side length c okay and if you if you're curious about this sort of thing you can do a little bit of poking around online you can you can look for for various proofs of the pythagorean theorem the sort of traditional proof i'm not going to get into the details of the proof but if you're at all curious um i think one of the ways that you would prove it is you would from this corner here you would drop a perpendicular on to the hypotenuse and and then you would make some arguments involving involving similar triangles um and so in fact this small triangle isn't is indeed similar to the original big triangle and with a bit of playing around and using the fact that for similar triangles side lengths have to be in proportion you can you can derive the pythagorean theorem our point here is not to try and derive the theorem if you're curious there there are lots of sites online that will do it for you but one thing that we should mention before we move on is that the pythagorean theorem is quite important for analytic geometry right for working on the cartesian plane and translating equations into graphs the pythagorean theorem is quite essential because the pythagorean theorem gives us a notion of distance so if you have two points in the plane so let's say you have a point here x1 y1 and somewhere else you have a point x2 y2 right and you want to know how far it is from point 1 to point 2 point a to point b you want to know that distance well you can construct a little right angled triangle where this side so we're putting that side that side right there's our length this side length here delta x is x2 minus x1 this side length here delta y y2 minus y1 right and the pythagorean theorem says that if you're interested in this distance d that d squared should be delta x squared plus delta y squared and so this gives you this familiar distance formula that you probably saw in high school that you take x2 minus x1 you square it you take y2 minus y1 square it add those together take the square root and that gives you the distance okay so that's where that distance formula comes from the distance formula comes from this pythagorean theorem which is really a statement about triangles now interestingly enough the distance formula comes from kind of understanding triangles sine cos tan they they're all defined in terms of these side length ratios from triangles despite that we're now going to pretty much forget about triangles and we're going to move on and we're going to okay so in the last video we introduced trigonometric functions in terms of ratios of side lengths for a right angled triangle and that's typically where most people first learn about the trigonometric functions um and and of course these these come up frequently they're they're useful in a lot of applications um used constantly in things like surveying right um but mathematically we tend to not really think so much in terms of triangles as we do in terms of circles and in particular this unit circle now so the unit circle is a set of all points in the x y plane that have length or distance one from the origin from zero zero now thanks to the pythagorean theorem thanks to the distance formula right we know that our our delta x is is just x our delta y is just y we want d to be one and and so we get a formula for this unit circle right so the unit circle is the set of all points in the plane satisfying this equation x squared plus y squared is equal to 1. okay and this angle theta that we see here is the angle between this radius that i've drawn and the positive x-axis okay so we can maybe pencil that in okay so it's that angle now you'll probably notice that just like we had here there is a right-angled triangle hiding in this picture if we drop this perpendicular down all right so there's the right angle triangle sitting there this side has length x this side has length y and so if you if you refer back to how we define sine cos tan in in this context here sine and cosine in particular as these ratios of side lengths well then right away you can see that x is going to be well x is the adjacent side right adjacent over hypotenuse which is just one so x over one is cos theta y opposite over hypotenuse is sine theta right now you can think of those as consequences of the right angle triangle definition but in fact when we're doing calculus we take these as definitions so so from the mathematical point of view this is the definition for sine and cosine as functions okay as functions of this angle theta now there's one thing that we have to we have to add okay sort of an important thing theta whenever we're doing calculus theta must be measured in radians so this is important you might be used to measuring angles in degrees and degrees are convenient for certain things um surveying perhaps they're they're nice when you're doing right angle triangle trigonometry but if you want to do calculus you have to work in radians um and and one of the reasons that that we want to work in radians is that if you work with radians it means that you can treat theta as a real number now why it's still an angle how do you get to treat theta as a real number well the reason is that for any angle theta that you pick there's going to be starting here at the point 1 0 there's going to be a little segment of the circle starting at 1 0 and ending at this point right like so let's call that s okay so s is this little sort of segment this arc okay so one of the basic formulas that you have from sort of circle geometry is that as long as you're working in radians that length s is just it's the radius times the angle so s is r times theta okay and because we're on the unit circle r is just what okay so s equals theta so that means that every angle gets identified with a length lengths are measured in using real numbers so what that means is that these these quantities here sine theta cos theta which were given before as ratios of lengths we can now treat them as well i mean they they were already always numbers but theta was this it was measured in degrees right so we don't you know when we're measuring theta in degrees or even in radians we're thinking it was an angle not as a number right but this gives us an identification that every angle produces a number right and and in terms of the quantity they're equal so so now we can think of these as functions we can think of the sine and cosine as functions and functions of a real variable just like we have exponential functions and logarithms and polynomial functions these are now functions functions of a real variable so we can do calculus with them that's kind of the the whole the whole point of moving to the unit circle is now we have functions of a real variable right we think of theta as a real number um we'll find later on once we get into doing calculus once we start talking about limits derivatives things like that if we want any of the the formulas that we derive to work out we definitely need to be working in radians if you if you work in degrees you're going to be off by by a factor of like pi over 180 something like that okay so it's essential that you you view theta as as measured in radians the other kind of nice thing that happens now is there's a lot more freedom in terms of angles that you can choose in fact theta is going to be a real number and and and we could in fact say it can be any real number this is a surprising thing right if you're here working with it with a right angled triangle then theta if you're measuring in degrees it's got to be somewhere between 0 and 90 degrees right it has to be it has to be an acute angle in this context right or if you're working in radians somewhere between zero and pi over two let's say okay so um but now you can you're not limited to acute angles right you can you can let this point go anywhere on the circle and you can still define sine and cosine for points right so sort of your first quadrant angles those are the ones that correspond to right angled trig but there's nothing stopping me from putting a point let's say out here right drawing that line my angle now goes around all the way around like that that's my theta but i still get a point on that circle that point still has x and y coordinates um there is i'm in the third quadrant now so they're both negative but i can still i can still assign those values or i can still figure out what the x coordinate is that's still going to be cos theta the y coordinate that's still going to be sine theta right and in fact i i can even go around more than once i can go 10 times around the circle right that's going to define an angle i can i can figure out what cos of that angle and sine of that angle is i can also go clockwise rather than counter clockwise and go the other way around the circle i can get negative angles right so now you're you have a lot more freedom in the sorts of angles that you can consider you can still define sine and cosine for those angles right now for any real number angle not just for angles measured in degrees between 0 and 90. all right um so let's dig a little bit more into this radian measure i'm just to go through the details there in case you're you're not too familiar with how do you measure angles in radians the key is to remember that radians are defined so that this arc length formula works right so that the length of a sector of a circle is just the angle that is spanned by that sector times the radius and since we're working with the radius of one right that means that a given length a given segment of the circle is just equal to the angle that's spanned so that means that one revolution in in radians is equal to the circumference okay and we know that circumference is given by two pi times the radius and our radius is one so the circumference is two pi okay so that means that one revolution is 2 pi radians which is as we know 360 degrees okay so all the way around starting starting here so by default zero is when you're you know we measure angles from the positive x-axis so here we're at zero one trip around we've gone through two pi radians so that means that at half a trip around we've gone pi okay pi radians okay a quarter trip is half of that so half of pi is pi over 2 right and so we started at zero if we go all the way around we're at we're at two pi right three quarters of a trip three quarters of two pi is is going to be three pi over 2 then you get to 2 pi and then you can subdivide further right so what you might have referred to as a 45 degree angle is now well it's half of a right angle so pi over 4. okay so this is a 90 degree angle right so a third of that a 30 degree angle would be we divide by that by 3 we get pi over six so pi over six is going to be somewhere around here okay on the other side you have pi over 3. in degrees that would be a 60 degree angle right so we have 30 45 60 90. right and now you can keep going around the circle so here you're going to have 2 pi over 3 also known as 120 degrees right then oops 3 pi over 4 then 5 pi over 6 right and then down to pi and you can continue those all the way down we can go the other of the direction here extend these two diameters like so seven pi over six this is going to be five pi over four this is going to be four pi over three three pi over two this is five pi over three seven pi over four and finally 11 pi over six right that completes your your unit circle all right now of course there are there are lots of other angles that you could mark uh the reason that i've gone with these ones is that these are the angles for which we can compute exact values for the sine and cosine functions right at least the ones that we can easily compute exact values for a lot of people get concerned because we've we've marked so many angles around the circle that you know oh my god i'm going to have to memorize all these things it's not so bad the only ones you really have to remember are the first quadrant angles and this is one of the nice things about working with radian measure okay is that if you look at the denominators it's going to tell you what is kind of the corresponding first quadrant angle right so if somebody hands me like 7 pi over 6 right i say okay 7 pi over 6 well i see a 6 on the bottom so the angle in the first quadrant that matches it is pi over six and or if i see like like seven pi over four or five three pi over four right these ones with fours on the bottom they they correspond to pi over four and it turns out that once you know the coordinates for these ones the other points are just reflections right so this point for 7 pi over 6 right it's opposite that one so if i know the coordinates for this point i know that in quadrant 3 i change the sign on both coordinates right both coordinates are negative down here so if i know that point i just put minus signs in front of the two coordinates and now i have that one right same thing for all these ones down here i just take these ones i put minus signs out front right if i'm in quadrant four i know that x is positive y is negative so again i reflect across right if i have the pi over three one switch the sign on the y coordinate and i have the i have the values for five pi over three right so so these these are the the basic ideas that you use right so you don't have to memorize the whole circle you don't even have to remember you know these like cast rule or anything like that you might have learned um as long as you remember in each of the four quadrants which coordinates are positive which are negative and you remember these values you can you can work things out for for any of these points on the unit circle the other thing that of course might come up is somebody gives you something like i don't know 31 pi over 6 and asks you to calculate sine or cosine of that angle and you're like well 31 that's that's not on my circle like what do i do with that right and so what you do is you have to kind of think in terms of of two pi's you have to say so 2 pi would be if i multiply by 6 over 6 that's 12 pi over over 6 right so this would be i could do 12 pi over 6 plus 12 pi over 6. i'm up to 24 and then i'd need 7 pi over six right so you can kind of break things down like that so so this is a full revolution right that's a two pi that's a two pi and so this angle is starting here going once around twice around and then around to seven pi over six right so if you if you're dealing with angles that are outside that range from zero to two pi you just simply add or subtract multiples of 2 pi until you get something that's one of these and then you can work out the answer so in the next video we'll tell you what some of those coordinates are so that you can start assigning values to the sine and cosine functions okay so in this video we're going to try to derive values for sine and cosine for these special first quadrant angles right now there there's there's a couple that we can get right away because we know that we know that for for any point right for any point on the circle with coordinates you know this point has coordinates x and y we know that x is cosine of my angle we know that y is sine of the angle that's how sine and cosine are defined right as the coordinates where theta is this angle measured from the positive x-axis all right so when theta is equal to zero we can see that we're at the point 1 0. so that means that of zero is one sine of zero is zero when theta is equal to pi over 2 well then you're at the point 0 1 right and so that means that cosine of pi over 2 is 0 and sine of pi over 2 is 1. and similarly the other four intercepts you can you can read off the answer right cosine of pi is minus 1 sine of pi is 0. cosine of 3 pi over 2 is 0. sine of 3 pi over 2 is negative 1. and and by the way for these the ones below the x-axis a lot of people will tend to measure those going clockwise so rather than 3 pi over 2 right it is minus pi over 2. you can do it that way as well all right now what about when theta is equal to pi over 4 okay well pi over 4 is right in the middle right it's this one that splits it in two and so there's some symmetry there you have the same amount on this side as you do on that side to say that in this case x is equal to y so we have a right angled triangle which is in fact an isosceles right angled triangle right and and we can work it out i mean the other way you know it's isosceles is that those two angles have to be the same right and and we know that for triangles the the angles also influence the ratios of the sides so if those two angles are the same the side lengths have to be the same we know that the hypotenuse has length one and we know from pythagoras that x squared plus x squared well that has to give me 1. so that means that 2x squared has to give me 1 right that means that x squared is one half i'm in the first quadrant so in the first quadrant x and y are both positive so i can take the positive square root and that puts me at the point 1 over root 2 1 over root 2. if you feel compelled to rationalize the denominator here go right ahead you could also write this as root 2 over 2. it really doesn't matter nobody in university is going to care if you rationalize your denominators so i wouldn't worry that much about it cos pi over 4 is one over root two sine of pi over four is one over root two um in fact one of the reasons that you have to rationalize your denominator um historically this was necessary because it used to be that you know you would calculate these values on you know as like a slide rule pre-calculator days and and so you could work out what root two was on your slide rule and then you could divide by two and that that you could do but you couldn't actually do one over root two that was not an operation that you could do so everyone had to learn to rationalize their denominators now we have calculators it's not really necessary anymore okay so you can leave it like that now pi over six so there's a there's a trick with pi over six which is you take you take your your triangle here and you double it so you you put the mirror image down below and one of the things you can work out is if this is if this angle is pi over 6 that's a pi over 2 right it's a right angle you can work out that the remaining one has to be pi over three right if you want to switch to degrees for a second right we know that the interior angles for a triangle have to add up to 180 degrees if we have 90 and 30 the leftover one has to be 60 degrees right so if we reflect that across symmetry says that this is also a 60 degree angle and if you look at the big triangle well two 30 degree angles add up to a 60 degree angle so this is an equilateral triangle right if all three angles are the same all three sides are the same and we know that this side has length one length one so that side has length one and that means that again because of symmetry those two sides have length one half because you take a side of length one and you're splitting it in two okay so now i know that i know that my y coordinate here is one half and i know that x squared plus y squared has to equal 1. so x squared so squaring a half gives me a quarter equals 1. so x squared if i subtract a quarter from both sides x squared is is three quarters so to take the square root you take the square root top and bottom pi over six is at coordinates x is root three over two y is one half okay so that means that cos of pi over six is root three over two sine of pi over six is one half okay very good all right so that's not so bad uh the only one that's left over is is pi over three but you can you can kind of use some symmetry here that if you you can exchange roles right um or you can go back to your right triangle trig right for for this angle uh opposite and adjacent kind of switch roles right so sine and cosine switch roles um and so you can work out that um the the last angle to deal with in the first quadrant pi over 3 well that's when x is one half and y is equal to root 3 over 2 and so that means that cos pi over 3 is a half sine of pi over 3 is root 3 over 2. very good okay so so those are the those are the basic sort of first quadrant values that you want to know when you're doing trig working with the unit circle pretty much everything else you're going to have to either rely on either trig identities or use your calculator to get values for other angles these are the only ones that are kind of easy to work out everything else takes a little bit more effort now the reason that i didn't bother with anything else here is that like i said all the other angles they're related to the ones in the first quadrant through some sort of reflection so if i go to something like 5 pi over 6 right well that is directly across from pi over 6 right y coordinates are the same x coordinate is opposite so if i know that this is at root 3 over 2 and one half i immediately know that this is at minus root 3 over 2 and one half right and so if i'm doing cosine of 5 pi over 6 i know it's minus root 3 over 2 if i'm doing sine of 5 pi over 6 i know it's one half right i'm just reflecting across so i change the sign of the x coordinate same thing if i'm down at say 7 pi over 4 right x is still positive so cosine of 7 pi over 4 is going to be 1 over root 2. sine of 7 pi over 4 is going to be minus 1 over root 2. right once you've got the first quadrant you know everything else you just have to pay attention to signs for the quadrant you're in remember that cos is always the x coordinate sign is always the y coordinate and you'll be okay okay so next we're going to look at the six the six trig functions some will refer to these as the six circular functions since since really we define them in terms of the unit circle okay so we have sine theta we have cosine theta and again remember that sine and cos they are defined in terms of the unit circle right so so they're the primary functions all right there's a bad unit circle okay so cosine is the x-coordinate sine is the y-coordinate on the unit circle so we have these primary ones so if we start thinking of these as functions of a real variable and maybe we'll think of them as sine x cos x is a little bit confusing because we have x y on the circle domain is is r and let me just for reference we're going to point out what are the zeros of the sine function the zeros are at 0 plus or minus pi plus or minus 2 pi and so on so any multiple of of pi cosine the domain is also r uh the zeros right so sine is equal to zero when when the y coordinate is zero so at the x intercepts cos is equal to zero at the y intercepts so at plus or minus pi over 2 plus or minus 3 pi over 2 plus or minus 5 pi over 2 and so on so all the odd multiples of pi okay now we'll get to tan theta so tan theta is defined as sine theta over cos theta one way that you might want to think about it it's it's really you know it's really slope okay it's the slope of this line segment right because it's y over x it's rise over run so tan is sort of measuring the slope and the reason that i mentioned the zeros for cosine is that of course tan is given by dividing sine by cos and that means that the domain the domain for tan is got to be well x can't equal plus or minus pi over 2 plus or minus 3 pi over 2. and and so on right it can't it can't equal the places where cos is zero but it's equal to zero at all the places where sine was equal to zero so at x is equal to 0 plus or minus pi plus or minus 2 pi and so on okay now those are the three main trig functions but you can also look at their reciprocals so there's also cosecant theta which is one over sine theta there's secant theta which is one over cosine theta and there's cotangent theta which is cos theta over sine theta which is the same thing as one over tan theta okay all right now a couple other things that we can we can say about these um sine theta and cos theta they're always between -1 and 1 right because they are coordinates on the unit circle right and the y range for the unit circle and the x range for the unit circle is minus one to one all right so so these two are what's called bounded um they're also they also have this property of being periodic right because once you go once around the circle you're back to where you started and the and the values start repeating um and that periodic property is is inherited by all six of the of the circular functions right but the other ones are not bounded all the other ones have vertical asymptotes right all these places where where these are undefined are vertical asymptotes for these functions right so cosecant is going to have vertical asymptotes at all the integer multiples of pi secant is going to have vertical asymptotes at all the odd multiples of pi over 2 right cotan is going to have vertical asymptotes at all the multiples of pi and so on right and and of course since these ones are always less than or equal to one in absolute value cosecant and secant are always bigger than or equal to one in absolute value so i i mentioned this because in the next video we're going to briefly look at graphs for these six so we're gonna change gears right we're going to rather than thinking of these as functions now of an angle we're going to think of them as functions of a real variable we're going to plot them in the cartesian plane so what we're kind of doing is you imagine as as you go out along the x-axis if you imagine that's your angle varying you're going to watch you know what happens to the x-coordinate what happens with the y-coordinate and you're going to plot those right and and this is going to generate uh the graphs for these so we're just going to give you a rough idea of what the graphs look like because they're going to come up and it's useful to have that picture in your head for these functions okay so we've got two videos left on trigonometry uh in those videos we're going to look at some of the identities that you might need to use on a day-to-day basis in calculus they come up less often than you might think so you don't have to worry too much about playing around with identities but they they do come up from time to time so it's something to be aware of now basic identities are some of these ones that we we observed when we were pulling up the graphs right we saw this sort of translation identity right that sine x is the same thing as cosine of of x minus pi over 2 okay so we have this sort of shift that that relates the sine and cosine graphs we have the sort of even odd identities let's label those so we have the even odd identities so sine minus x is minus sine x cos of minus x is plus cos x right so sine is odd cosine is is even among the other four trig functions secant is the only other even one uh the other three uh tan cotan cosecant they're all odd as well okay and then we also have these identities coming from the fact that the trig functions are all periodic so so sine of x plus any multiple of 2 pi is the same thing as sine of x for k k could be any any integer so zero one two three minus one minus two minus three uh and the same thing for cosine adding any multiple of 2 pi gets you back to where you started same is true for secant and cosecant for tangent and cotangent you'll notice that in fact the the period is a little bit shorter um the period for tan is in fact just pi okay and that's valid for any for any x right for all these identities they hold true for for any angle x that you want to put in now the fundamental identities there are the pythagorean identities so-called because well they come from the pythagorean theorem remembering that that sine and cosine are the x and y coordinates on the unit circle and remembering that the unit circle is defined as as x squared plus y squared equals 1 we get that cos squared x plus sine squared x is equal to 1. okay so that's this primary pythagorean identity all right tends to be the one trig identity that everyone remembers the the ones that are a little bit tricky to derive we're not going to try to prove the addition identities and and i guess also subtraction so the these addition subtraction identities they're they're the most difficult trig identities to um to derive and they're probably also the most difficult to remember so for cosine cosine of x plus y is cos x cos y and then it's the opposite sign minus sign sine x sine y if you were doing subtraction if it was x minus y that minus there becomes a plus okay and for sine sine of x plus y you get sine x cos y same sign plus cos x sine y okay and if it was a minus sign here it'll be a minus sign there okay um so these these are these three entities here are kind of your go-to identities these basic identities they'll come up from time to time but but these ones you kind of tend to internalize and you don't think about them too much maybe this shift one you don't remember which way it goes for the shift one isn't isn't super important anyway these are going to be the primary ones that you rely on quite frequently and hopefully you'll use them often enough that you don't have to sit down and memorize them because they'll they'll sink in once you've used them you know enough times okay um so these ironies here i've given to you as the fundamental identities because pretty much every other trig identity you can think of can be derived from these right so everything else is is derivative of these ones um i'll list a few of the most common identities that you can derive from these i'm not going to do them all because there's there's you know you can come up with with hundreds of identities you know you just play around you can come up with all kinds of identities right in um you might have you might have done a unit on on this in high school where you spend time you're given all sorts of different identities and you're asked to show the left-hand side equals the right-hand side you probably won't have to do very much of that in your calculus course but you might need to make use of identities to simplify certain problems right you might have equations with trig functions in them that you need to solve or you're just trying to simplify things because it makes it easier to evaluate a derivative or an integral or something like that so some of the ones that you're going to run into there are the ones that are derived from the pythagorean identity that give you relationships between the other trig functions right so if you if you take the pythagorean identity and you divide everything by by cosine you get well cos squared over cos squared you'll get 1. sine squared over a cos squared you get tan squared okay and then 1 over cos squared well 1 over cos is secant squared so you get secant squared x similarly you could divide everything by sine squared co squared over sine squared gives you cotan squared sine squared over sine squared is one and one over a sine squared is cosecant squared okay you probably won't see these too much in calc 1. once you get to calc 2 when you're doing some techniques of integration you're looking at trig substitution and things like that for integrals you're going to see these quite a bit they're going to pop up all over the place okay the ones that show up throughout calculus quite frequently are these double angle and half angle identities that you can derive from the addition formulas usually by say setting x equal to y or something like that so we have ones like if we do sine 2x well keeping in mind that sine 2x right what is 2x 2x is just x plus x right um so if you put x equal to y in in the identity for sine of x plus y you're going to get sine x cos x plus cos x sine x so you just get sine x times cos x twice so what you get is 2 sine x cos x and this identity is a good reminder that you can't just take that 2 and bring it out front there's a lot of people that are always tempted to do that especially once you get to things like limits involving trig functions a lot of people want to bring that 2 out the 2 doesn't come out it's stuck inside the sine function if you want to bring it out you can but it's going to cost you a cos x and for other multiples 3x 4x5x it gets a lot more complicated you can keep using this addition formula repeatedly and keep expanding expanding expanding and get more and more complicated formulas but generally that turns out to not be all that worth your time okay now using that same idea in the identity for cosine we're going to get cos x times cos x so cos squared x minus sine x times sine x so cos squared x minus sine squared x right with a plus sign you just get 1. with a minus sign it's not quite as simple you get cos 2x cos squared minus sine squared same thing as cos2x but there there are a number of different ways that you can sometimes write this identity right so if you write this co-squared as as 1 minus sine squared if you plug that in right solving for cos squared there i get 1 minus sine squared if i substitute that in another way to write cos2x is 1 minus 2 sine squared x if instead i leave the cos alone and i say that sine squared is is 1 minus cos squared and i remember to push the minus sign through the brackets then i can also write this as 2 cos squared x minus 1. and these can be useful because these lead to these so-called power reduction formulas there are times where you're dealing with a an odd power or sorry an even power of sine or cosine you got a sine squared or a cos squared in some equation you want to you want to lower things down a bit uh especially when you're getting to integration in integrating even powers of trig functions the only way you can deal with them is by playing around with some identities so if you take this equation here cos 2x equals 1 minus 2 sine squared and you solve for sine squared you get that sine squared x is going to be so i'm going to move the sine squared to that side move the coast to that side 1 minus cos 2x i've got to divide by that 2. similarly cos squared x right if i move the 1 to the other side cos 1 plus cos2x divided by 2 1 plus cos 2x over 2. okay um so these these so-called power reduction formulas those come up again when you're getting to integration problems you'll use those quite frequently they they come up more often than you might think the other place where you might use them is simply that you're trying to calculate let's say you know you know sine of pi over 6 and you're trying to come up with sine of let's say pi over 12. well if x is equal to pi over 12 then i would have a cosine of pi over 6 over here which i know right and i can take the square root of both sides to work out the value for for sine of pi over 12. so sometimes you're using these identities just to get values of sine and cosine for angles that are not those standard angles on the unit circle they can come in handy for that as well probably most frequently where you're going to see these is is once you get into integration you might have to do these to simplify integrals in calc 1 maybe you're going to use them to try and simplify some equations involving trig functions like you're trying to figure out where the derivative is equal to zero maybe it's going to come in handy to be able to write sine 2x as 2 sine x cos x to try and simplify an equation you might see something like that as well that's uh that's it for our our precalculus review material from here we're going to move on to calculus videos

Channel: Geek's Lesson

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Keywords: precalculus, precalculus full course, precalculus functions, precalculus algebra, precalculus trigonometry, precalculus for beginners, precalculus for dummies

Id: 9IOTI9ViExg

Length: 425min 2sec (25502 seconds)

Published: Sun Dec 13 2020