College Algebra - Lecture 9 - Functions and Their Graphs

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well we're on to another unit this time it'll be functions and their graphs now that we've seen how to graph a few things I think you'll find this very interesting section we'll learn about the key idea of function which is one of the most important ideas of mathematics and then we'll learn how to graph functions in various ways and then at the end we'll do a number of problems that are real-world problems you might in the past have called them word problems but these are real-world problems that actually allow you to construct functions and their graphs and see how they apply on that note we'll go on functions and their graphs all right the very first thing of course is that we need to define what functions are function is really the central idea of mathematics took a long time to develop but we finally have it and we can use it in all of the material that we're going to use for the rest of the course so on that note let's start talking about what a function is all right the function idea well first of all let me make a little note here that function is being used in a technical sense that is to say I am going to define what function means in mathematics now function means a great deal of things in in normal everyday language and some of those actually apply to what we talked about but there will be a technical definition that I'm about to give you okay first of all I will give you what I might call two visions of function two ways to view the same idea here is one way and either one of them is fine if you put both of them together then you get a better sense of what a function actually is the first way is as a correspondence I'll need to explain this with examples don't worry those are coming up but a correspondence between two sets and the two sets will be given special names also so that's one way to think of a function another way I'll put it or here another way that is also fruitful and gives you a different viewpoint is to think of a function as a machine again I'll need to define what that means a machine with input things going in and output things that come out now with these two ideas you can put together I hope in your own mind what a function is so let's start with the first one a correspondence between two sets and then we'll go on to the second one afterwards so the first one number one is function as correspondence sometimes this is referred to as a is a one-to-one correspondence but a one-to-one correspondence really has a specific meaning that I I'm not going to be using here so I want to be careful and just use the generic word correspondence and here's the basic idea imagine that you have a set and for no good reason let's just call it set X over here on the left and here it is it's some set of real numbers now the sets we look at in this course of course are real numbers except for certain cases when we look at complex numbers and those will be a little later and let us suppose that you have a second set a set that we'll call Y and it's over here somewhere okay now in the set X there are various values okay let's suppose as an infinite number because those are the kinds of sets we'll be looking at and over here and why there are also values probably an infinite number and what the function is the function is actually a correspondence that says say this point is related to this point so this is the correspondence here for this point and this point so they become a pair they are linked we'll have to explain exactly what the linkage is but you can see that we can develop a correspondence where each point here is associated with some point perhaps in the other set and this would continue until all of the points over here are associated with points over there now again remember both of these sets are all real numbers that remember is our symbol for real numbers we're talking about real numbers and only real numbers for the time being okay I need to be a little more specific here there are some things that need to be carefully defined when we're thinking of a function a correspondent imagine that you're over here living in set X and that you want to get on an airplane say or a car and you want to travel over to the other set now the correspondence is such that it is always the same so that if you get into this airplane or car on one day you will travel to this point if you get into it on another day you will travel to the same point it was always the same thing the function the correspondence is always identically the same now of course it makes sense and is possible that two people over here person there and a person there might end up going to the same place so there might be an arrow that let me pick another air I haven't used this arrow goes to here and from this point over here this person could also go to the same destination now there's nothing that makes any there's no difficulty with that two people can go to the same definition destination what you don't want to happen is that one person on different days at different times would go the correspondence would send it to a different point all right now that's a bit vague let's go ahead and write down specifically what I mean by that so this is the first version of the definition of a function this will be the correspondence version a function and of course once I get writing I'll start abbreviating function as I abbreviate everything else but for now I'll write it out a function is in this vision a correspondence from the set X into the set Y okay those will have specific names later that what does this correspondence do it associates with each point in the X set with each and let's say little X's live in big X so little X lives in big X so with each little X and big X some and this is a very important word now unique some unique Y in big Y ok so every point here in the X set is associated with if you like every X over here goes by airplane or correspondence to a unique destination Y in Big Y ok so there's only one place it goes to its unique so another way to see that is with the arrows that I had on the other page this means the following situation is OK that if you have two of these elements that live in the X set say column x1 and x2 to distinguish them it is possible that they can both go to the same element Y in the Y set that's ok what is not okay is the other arrangement where you have a single X say x1 again going to two different Y values where these are not the same this cannot happen if you have a function we want function to mean this or going to one and this one going to another one but never this and of course we don't want x1 to go to three places or four or any of that so another way that people often say this is this is many to one and this would be one too many so many to one is ok but one to many is not ok so let me repeat a bit of that and write some of that down so you'll have that in front of you again putting this in words each X each little X and and you heard me use these this terminology people do say this goes to or corresponds to or is associated with each X goes to only one only one y ok another way to say that as I said before is there's no ambiguity about the destination if you think of it in terms of planes or trains or cars alright so there's no ambiguity in about the destination now that's a nice colloquial way of saying this now let me give you a specific example here's an example here is the function I want to look at and we haven't got function notation yet so you'll have to bear with me if you already know it we want the function that takes little X's and I'll use just an arrow like this and it takes it to squares okay so every little X is taken to the corresponding square so the set of all the little X's that are allowed in would be the big x set and the set of all the squares that come out of that would be the big Y set so big X in this case can be if I allow it to be as big as possible it can be anything because I can start with anything and I can square any real number so imagine this is my set of real numbers over here and then over here my bid Y well based on my own experience squares can only be 0 or positive they can never be negative so I know that Y if it's allowed to be as big as possible would be 0 to infinity it would include everything possibly out to infinity all right here's a symbol for that now the correspondence is via x squared so let me take some specific numbers over here here's a number minus 2 now it's going to correspond to what well it's going to correspond to 4 why because the correspondence here is square take -2 square it and you get 4 now notice in this case because of the nature of this function 2 has the same destination because 2 squared is also 4 and then you can put in other numbers one-third of course would go to 1/9 because minus our minus one-third would go to 1/9 minus 1/3 squared is 1/9 zero would go to itself because 0 is equal to 0 squared and etc there's an infinite number of real numbers here and there'll be an infinite number of elements over there so there's an example of a particular function and what really is happening is that this correspondence sort of lives here in the arrows the arrows are the correspondence if you imagine your hand going in here and gripping this say for example in this picture this is the gripping hand picture of a function okay this is a function as correspondence the function has correspondence it's like a big hand that grips this and you give that fist a name another way to look at that might be this the function is this correspondence we've grabbed it all together in this in this ring and in fact this is a specific function this is a function that I've marked with the question mark because it's a function we won't cover in algebra but zero under this function goes to zero pi goes to minus 1 PI over 6 goes to 1/2 2 goes to approximately 0.909 and then there's an infinite number of real numbers that can go in and that will come out now some of you out there may be able to guess what this function is but don't worry about it the important thing is that you have an idea that a function is a correspondence of some kind between here and here as long as each value on the left is always going to go to the same value on the right in fact if you put those values together a final note on the correspondence version here a function as correspondence which is what we've been looking at in all those pictures with more examples to come Geils a table now you must have seen tables before we'll set up this table as follows here on the left will be set big X here on the right will be set big Y and what will be the line in between that will represent the function let's do the x squared function again so we'll put in say minus 2 and of course we get out 4 now we have a table with numbers on either side we put in 1 we get out 1 we put in that minus 1/3 we get 1/9 let's put in PI for fun we'll get out pi squared and then an infinite number of numbers on either side so what happens with thinking of a function is a correspondent as you can think of it as a table in other words if you know where the number begins here as minus 2 and you know the end result and you know that for every number in here then you know the function because you know it by what it does even if you didn't have the formula x squared if you had all of these numbers and all of those numbers and you had them paired like this you would know what the function is now that's going to have bearing when we start wondering about how to graph a function so let's go on to our second vision of functions the machine version to function as machine alright function as machine well this gives me a chance to draw my standard machine for a function it's going to be very generic now all I need for my machine to have is an input place and an output place so I'll even give it a little 3d for you all right there's my function me jazz it up here with a little third dimension there we go I'm having fun here all right there's a little third dimension okay this is the function okay this is the function whatever the function is as a machine and what goes in over here this is where the input comes from and the whole set of input elements would be the input set and they will go into the function what will come out the output set the output set will come out and form the output set input set output set function in the middle now you can see how this is related to the correspondence idea the function as machine simply says the correspondence is now going to be considered a process alright you can put that down if you like instead of being a static correspondence where you have the input set and the output set paired as in that table you can now think of it as a moving thing the input elements go in the function acts upon them and produces the output so we have this machine version of the definition this can be very pleasant I'll show you some examples later so let me write out the definition I wrote out a definition before let me write it out again this time we'll call it the alternate definition of a function and here's how it goes try and make it sound similar to the other one a function is a machine in this incarnation that associates with each and this time we'll call them input elements with each input element I'm abbreviating element right there with ELT some unique same see this should sound just like the other one so the associates with each input elements some unique output element now previously we call those the little x value out of big X and the little Y value out of big Y but now I'm calling them input and output because that makes more sense in the Machine context again there's no ambiguity each input X value if you like always has the same output the same output doesn't matter when you do it you will always have the same output there is no ambiguity that is what makes a function see with a machine it seems even more sensible because when you put something in you expect the same thing to come out time after time after time you don't expect to put something in and to get out something one time and something else another time so each input always has the same output so let's go ahead and look at the example we looked at before so as before we looked at the function that took X 2 x squared okay we don't have any standard notation for that yet we will but that's the function that squares things now in this machine version there's the standard machine again okay I just need room for input and output now the function in here is the square function and actually you know writing x squared seems like the right thing to do but I think there's a better way to do this I'm going to write instead of this I'm going to write box squared and you'll see why later I'm just going to write it's a thing squared now here are all the things coming in -2 is going to go in here 1/3 will go in here 0 will go in here and say 5 will go in here and of course there's an infinite number of numbers that can go in there what will come out of the machine well this is the square machine the -2 goes into the Box gets squared and comes out as for the 1/3 goes in to the Box gets squared and comes out as 1/9 the zero goes into the box gets squared and comes out as itself zero the five goes into the box gets squared and comes out as twenty five five squared and of course the infinite number of numbers going in will result in an infinite number of numbers coming out so there's one way to visualize this function again this side over here is the input side and this side over here is the output side okay now using this machine imagery I'm going to go ahead and give you what I like to call a sampler of functions now some of these functions we will cover later in this course there's one of them that we won't cover in this course at all but you'll see if you go on and take something like calculus alright now this is a function that's rather strange it's unlike the function x squared say that we looked at before this is a function that's in pieces all right this is a function in pieces and my pens seem to be drying out on me here so bear with me as I switch pens all right a function that's in pieces now here's how I'm going to define it I'm going to say that this function gives me minus X for X's that are less than or equal to zero let's say it gives me five always five for X's that are greater than zero and less than two and let's say it gives me something more complicated like x squared minus one for X's that are greater than or equal to two so let me encase this in the machine image that I had before and let's see what it does to a few numbers now we're going to see functions like this but we we can look at them even now say minus two one three halves and one more say three and of course there's an infinite numbers a number of numbers that can go in let's see what happens - - now that is a number that is less than or equal to zero so four numbers that are less than or equal to zero we take - the number - two negated becomes two one we look in here and where does one live one lives between zero and two so four numbers that are between 0 & 2 the output is always 5 it doesn't matter what the number is so if that one gives me 5 what about 3 halves well 3 halfs also lives between 0 & 2 so again the output is 5 so there are two numbers that go to the same place under this function which is ok as far as our definition goes and finally 3 let's see 3 is bigger than or equal to 2 so 3 has to be at the output of 3 is what results from squaring 3 which is 9 and subtracting 1 and from that we get 8 so there's an example of a function we will explore these kinds of functions later but I think you see that I've got the machine here clearly defined and the numbers that go in we can tell exactly what's going to happen to them as they come out of course it would be much harder if I didn't tell you what the function was we won't do that not for now anyway all right here's another one now this is a function that we will discuss later in this course 5 to the X power so let's take a few numbers into that just to see how this whole correspondence or machine version of the function might look ok so I'm picking a few numbers here let's put 0 in 0 goes into the position of X then 5 to the 0 of course you know is 1 so 0 is associated with 1 by this function - 2 goes into here that's 5 to the minus 2 power now if you remember what that is that's 1 over 5 squared which of course is 125th so minus 2 is associated with 125th under this function pi goes in and becomes five to the PI now what that equals we don't have a clue at this point but that's what happens when that domain element that element on the left goes in for x3 going in for five becomes five cubed and of course 5 cubed is 125 and then we continue so there is another example of a function it's fairly standard function and we will see it later in this course then there are lots of functions that will remain undefined within this course for example trigonometric functions because that's part of a trigonometry course but here's a function I wanted to show you that I think you'll follow and it really is an extension of algebra but you don't get to see these things unless you take calculus to of all things and here is the function I'm going to put in here 1 plus X plus x squared plus X cubed plus dot dot dot all the way out to infinity now there's not too much we can do with this at this point but we can't evaluate this for a few numbers and I'll be using information that I have that you don't have but I want you to see that functions come in different forms we've seen ones that were broken up into pieces we've seen ones that were a single formula we saw one that was an exponentiation five to the X and now we see one that looks like a polynomial that goes on forever now this actually has an official name this is referred to as a series and you do study this as I said in calculus too now let's see where these numbers go if we put one in here I think even you can see that one plus one plus one plus one etc is going to go to what we must call undefined because adding up an infinite number of ones is not going to yield a real number the only numbers that we deal with now are real so whatever this is this is undefined for us you should also be able to see that when I put a 0 in everything with an X in it goes to 0 and you're left with 1 so there's another one we can do without too much trouble now putting in a 1/2 or a minus 3/10 if you had a little more knowledge you would see that 1/2 will go to 2 and minus 3/10 will go to 10 13 etc now that those go to these two numbers on the Left go to these two numbers on the right because I happen to know something about this series that you don't but the point is this is a legitimate function even though it doesn't look like other functions you've seen before all right I think we've looked at the pictures of functions enough we're going to stop now and then when we come back we'll look at language and notation of functions the language and notation of functions all right under the language of notation of functions I wanted to mention a couple of things up front these are just pieces of advice there are many versions of the names let's say of functions of the sets involved with functions of the correspondences there are many of them get used to them the function notation has developed over centuries and so there are many different versions that all mean similar things and you just have to get used to them because you never know which one's going to turn up and practice also the notation is new to you so practice it and practice it now while we're doing this section because if you don't you'll find yourself falling back into patterns of older notation that is that looks like this notation and you'll be wrong in those cases so let's go back to how we define functions we had on the one hand a set on the left that we called big X temporarily a set on the right that we call big Y and in the middle is where the function was represented as a correspondence or as a machine and now under each of these let me draw lines under each of these I'm going to give you the standard terminology each of these sets has a name the function as in its various incarnations will have different names so let me start with the basic ones the set on the left the input set is called the domain of the function that is the traditional name the output set is called the range of the function so those are the two standard names for the things going in and the things coming out of a function now the function itself can be denoted many ways it can be noted with single letters or abbreviations now that covers most of what we'll need to look at there are other ways to indicate functions and I'll show you that in a moment what else can we use as terminology from the left to the right now the domain let me rewrite this so that you'll know you have on paper what I said this is the input set and the range is the output set what wood elements within the domain be called and what wood elements within the range be called well you might think they're called domain elements and range elements and certainly they are but there are other names that are also used again because of history so the elements let's start with the domain of course as I said they can be called domain elements and I'll abbreviate element here domain elements they are also referred to as independent variable now this is in the context from the hit theory of equations and that's where we have variables x and y and one of them is independent one of them is dependent the of the value that goes into the function we will refer to as the independent variable and sometimes the value that goes into the function is called the argument okay that's a bit of an older piece of terminology but it's also used is the argument of the function then let's go over here and look at elements within the range set now of course as I said they can be called range elements which you might expect and the variable that comes out is referred to as the dependent variable dependent now that makes sense when you think about it because it depends on what the independent variable was the independent variable goes into the function and produces the dependent variable then the elements that come out are often referred to as the image elements the image under under the function the x value goes into the function and produces an image or sometimes they're referred to as the value of the function the value of the function so an X goes into the function and then the function produces the value which we can call the image we can call the dependent variable or the range element all of these pieces of terminology for elements in the range are used just as all of these are used for elements in the domain all right well those are the words that are used and now we really see from this picture that a function is in three parts so let me write that down and then we'll come back come back and we'll look at these three parts again and start and writing down some of the notation that's used alright so a function and this is one thing that you want to keep in mind a function consists of three parts now students tend to forget this they can tend to think of it only consists of one part let me show you what the three parts are first part is a domain and remember that's the input set why don't I write that down the input set the things upon which the function is allowed to act it is also consisting of a correspondence or if you like machine what the function actually does and finally a range which is the output set output set now as I said students tend to only look at the middle part and talk about the function as though it were simply a machine or a correspondence but the fact is the correspondence of the machine doesn't have any meaning without a domain set to act upon and without the results of that act action so function really does consist of three different things and the more you keep that in mind the better it will be for you okay all right now let me go back to that three-way picture that I had before so I had over here now I'm going to just call it domain and in the middle I had function and on the right I had range and let me divide it in three again and now let's talk about the notation that's used for this not just the words that are used the language but the notation domain elements are often just listed by single variables there are some traditional ones that we will see very often in this course of course X is one T is another one that we'll see fairly often because T often stands for time in problems where time makes a difference let me go ahead then go to the function how do we relate how do we notate or write down functions well as I said before we use single letters or abbreviations what do I mean by that well a function might be denoted by F or G or H or any of a number of other letters depending on the context if you're doing a problem in which the function you're looking at is an area function you might want to call it the function a so you pick a letter that's appropriate but when we're doing things in a generic manner will just use letters like these also there are some functions that have specific names the natural log function which is abbreviated by Ln log function which usually means log base 10 but can mean other logs exp which means the natural exponential function Si and which you may have encountered in your life which stands for the sine function or cos which stands for the cosine function etc there are lots and lots of familiar three-letter or two-letter abbreviations or more that are used to indicate functions but of course you can use any abbreviation that you like ranges what do the range elements look at look like well this is going to I'm going to talk about this in more detail on the next page but I'll list a few things here so you'll see what we're talking about the function here has to apply to the domain element to produce the range element the range element it would be best if we had a notation that told us what the function was and what it acted upon sort of a history of where the range element came from well here's the notation we're going to use and it'll require a little more explanation but I'll write it down suppose I use F here for the function and let me use X here for the domain element then the range element would be written as f of X this is pronounced f of X it's also sometimes pronounced f at X the notation is designed to give you that history I spoke of it tells you the function and what the function acted upon so you have a representation of the range element whatever the range element is and that tells you where it came from and how it got there what about some of these other functions here well you might have sine of X or you might have log of X I hope you're beginning to see a pattern here you might have Ln of X you might have exp of X you might have cosine of X now there's a pattern there the three letters are going with a pair of parenthesis in each case or the two letters in this case up here the single letter with a pair of parenthesis that's worth separating out so I'm going to do that right now that that new range notation is new enough that new range notation is new enough that it needs a bit of description one thing I want to point out is note f of X is not multiplication and when you've seen parentheses before in this course and F like this you might have thought this is the letter F times the letter X in which case the parentheses really seem to be superfluous but this is not multiplication what this is the actual correct notation would be f of blank these are the parts that go together the letter that names the function or the triple of letters if it's log or exponent cetera and the two parentheses they're all part of one thing be nice if they were all kind of linked together but they're not so you have to think of them that way and then what goes in here I'll put a box here to indicate it is the domain element goes in there okay and this is not F times anything it's f of something so when we write this example here for we write something like this you will see you might see f of x equals say 6 minus x over x plus 2 now that's what you're seeing now how do you read this you want to understand what this means x is the domain element the argument the independent variable all the same words all the same the same object different words what is coming out here of course this is the range element or if you like the image of X under F and F of course is the function so we have the function the domain element element in the range element the deranged element is written with this notation you know a better way to write this so that you see the process and don't get caught up on the letter here X is to rewrite this as f of box equals 6 minus box over box plus 2 now this makes this look a lot more like a machine it's a machine into which you put numbers and out of which you get this where the number that you put in 6 minus that number over that number plus 2 yields and output so I'm going to use the boxes whenever I can so that you don't get confused with the letters the box will literally indicate the function ok this is a way of describing the function so if you want to think of a function think of this notation ok f with its two parentheses and I put little dots indicate to indicate that they're linked the two parentheses are linked ok what goes in here this is where the domain element goes ok but the function itself is the larger notation I hope that image stays with you it's really big isn't it ok let me go ahead and show you now another example and I'll use that function notation and I'll write it out as a machine because we just talked about it that way I'll use a different machine for the sake of variety so here's a machine okay now this time why don't I use the letter T just to be different t is the input what will be the output I don't know because I haven't indicated what the machine is the machine function let us say it is going to be called G of box generic name and what will it be it will be box squared minus 2 times box plus 5 so that is what is this what this machine actually does it'll take T in do this to it and produce well let's see it'll produce T squared minus 2t plus 5 so T goes in T squared minus 2t plus 5 comes out and this is the function in between that generates that and this end result now is naturally called G of T the T has been put in the box and we have G of T so that's how this notation works and if you keep these pictures in mind you'll you'll probably be better off as time goes on let me bring that other one back remember big f parentheses I just like this picture okay now there's a terminology used throughout and let me rewrite that previous function I put G of box down do a box as box squared minus 2 times box plus 5 the fact is in practice you're not going to see boxes what you'll see is particular letters you might see G of x equals x squared minus 2x plus 5 or you might see G of T is T squared minus 2t plus 5 or you might see G of U I hope you're catching on to this G minus 2 u plus v dot dot dot all of these letters don't matter they're placeholders they're what we call previously dummy variables that's why I like using the Box placeholder what letter you use is irrelevant what matters is the structure of the function and it is identical in each case okay some other pieces of terminology sometimes we say sometimes we say in quotes the function f okay people will say that or sometimes they'll write or say the function y equals f of X so there's a couple of things going on here they're saying it is the function f applied to X and that's the range element but for short I like to call this Y because that's going to remind me of the Y values that occur when you graph points the X Y values will recall so very often you'll see the function written is y equals f of X and the Y is a shorthand for the full version of the f of X so let me put that down shorthand now you'll see more of this later when we start talking about graphs okay for example that leads me to another piece of terminology I can use what I have sometimes we know a function explicitly explicitly that means we actually know how to compute it for example y equals 3x squared plus 4 if I say how do i compute this for X equal 5 well I know that I square 5 x 3 and add 4 I have an explicit description of the function other times and again we'll see more of this later I'm just introducing the terminology other times a function say Y is only given implicitly now as you go on in mathematics you'll see more and more of these implicitly for example maybe given I have an example here something like this x squared minus XY plus y cubed equals 2 now you say that looks like an equation yes it is an equation but if I were to solve this for y I would get it into a form like it is up here and I could think of it thinking of the Y and the right hand side as some function of X I could rewrite it this way in which case f of X would be given explicitly however in this one although you can solve for y if you think about it for a moment you realize that it involves solving a cubic and Y when a function is given implicitly it may be very hard it may be hard or even impossible to a write such a function explicitly so you sort of have to be grateful for what you have it's an implicit representation of a function it is not explicit but I wanted to point out the difference in those two words and the fact that that terminology is used all right one last picture there's my function again f2 parentheses could be G could be Ln could be si n could be exp but the point is it goes with the two parentheses right it's all tied together sewed together if you will with these dots and in here goes the domain element and all that note will stop and move on to the next bullet now let's talk a little bit more about domains more on domains the first thing I want to say is that let's make our an agreement you and I we will agree that whenever a function is given to us a function that may have no specification we will assume that the domain is as large as it possibly can be so let me write that down unless specified because sometimes you'll have a function where the domain will be specified by the circumstances we agree okay you and I will agree that the domain as I just said the domain of the given function is the largest set here's the way we'll say this the largest set in the real numbers for which well let's see the largest set for which the function makes sense for which the function is defined for which f of X the range elements that come out of this are real numbers because right now we're only dealing with real numbers and you can put in there make sense if you like so I'll see if that reads alright unless specified we agree that the domain of the given function is the largest set in the real numbers for which f of X I guess we need to make that singular don't we is a real number okay which f of X is a real number or in other words make sense okay that's the first note about domains that I wanted to pass on and let me show you what I mean by that by giving you an example here's a simple example suppose the function and it's often going to be given just this way f of X equal 1 over X now you're probably going to want to think about that as F of box is 1 over box so that you don't get confused by the letter but f of X equal 1 over X say is the given function and nothing else is set so the assumption is the agreement is that we will assume that the X's that are allowed into the F in other words the domain of the function will be all the X's which make the range make sense make it a real number now you can see from the range there's only one number that will not be allowed that will be the x equals zero number so we agree that the domain of F is all real numbers except X equals zero now you know after writing all of that down we certainly know what the domain of this function is the domain that we are assuming is is the largest possible except that it took me a lot of words to write that down didn't it it would be nice if I had a notation that could make all of this a lot simpler say the domain of F we ought to have a notation to just to say that and then I'd like to be able to say all real numbers except x equals zero in a more compact fashion so I think I'll go ahead and introduce something like that to you right now and we'll call that point two here I had point one let's call this point two we're continuing with more on domains point two a shorthand notation for the domain of F for the phrase the domain of F is do M of F or sometimes it's just do m and people write it that way with do M F now I will use that from time to time because it's helpful it's certainly short and I think everyone understands what it means it means the domain set that goes along with the function f so it's simply useful okay and just for the record the range is indicated by our a n of F now we're going to spend much less time calculating ranges i mark that down just for completeness okay the third point under domains we'd like a compact way to specify domains because previously I just wrote it out now we actually do have some compact notations we have the notations for intervals so like intervals say and you remember the notation for intervals an example of that might be the interval the open interval from minus 3 to 1 or maybe the interval that's closed at four and goes off to infinity etc now that's a nice compact notation but what if your domain is not just an interval what if it's a combination of intervals or is some other set of numbers what we need is we create something that I left out of the basics part of this course because we didn't need it until now we create what we'll call set builder notation set builder notation it's a way of constructing a set that is generic it will be used here for domains occasionally for ranges and for other things later on so it's good to introduce it and what I'll do is I'll introduce it via an example suppose I want to take this set which I have to write out first because I don't have the notation yet the set of all real numbers or all reals say greater than five or less than or equal to say minus three now that is a set that if you think about it for a moment the number is greater than five and those less than or equal to minus three are not in a single interval in fact there are two intervals involved here so this set I will denote as follows and I'll explain the notation as I go the set will be indicated by putting curly brackets at each end those are also called braces but we'll go ahead and put one at each end and then the set will be first thing I will write down and the letter here again is arbitrary so the the numbers X in the real numbers where this little epsilon is a Greek letter e it stands for element of so we can read this is is an element of so x is an element of the real numbers this vertical line is read such that and what it is of course is a vertical bar if you like we have a vertical bar so we want a set of all the real numbers X and X and the real numbers such that what such that these conditions are met we can write those down as saying X must be greater than 5 or write the word or in there X is less than or equal to minus 3 so there is my fairly compact notation for the sentence up here in English and as these sentences the describe domains become more complicated you'll begin to appreciate this notation it is also completely standard in mathematics so let me read it again this is the set of all X's in the real numbers such that so here we're describing the kinds of numbers involved and the such that gives us the condition the actual description in this case X greater than 5 or X less than or equal to minus 3 so if X is in either one of these intervals then X is in this set which I tried to describe above so as we use this notation let me go ahead and show you how it will work example the domain of the function 1 over X can be written as the set of all the real numbers is such that X is not equal to zero now that's the example we had before this is the function f of X equals 1 over X that we had up there a moment ago so I'm using that notation the shorthand to the main event 1 over X is this set the set of real numbers such that X isn't 0 so everything that is not 0 can be put into this function all right let me show you another one the domain of the function square root of x another familiar function is the set of all the X's in the real numbers such that now what's the condition here to take the square root or in fact any even root of a number the number inside must not be negative so X and the real numbers such that X is either 0 or positive we can say X is greater than or equal to 0 now this one turns out to be a single interval so we can actually write it in interval notation which is even more compact but you see this works in all of these cases okay well these two examples actually point out a couple of good points about what's usually missing in a domain so let's go ahead and indicate what that is what's usually missing from a domain from a potential domain I guess I should say of course the real numbers is the largest possible potential domain what would be usually what would usually be missing well we're division by zero might occur now that was that as in that case for the function 1 over X we have to throw the points out the x values out that would make that division by 0 occur in that case it's simply x equals 0 another possibility is where even roots even roots of negatives or negative numbers real numbers of course for us negative numbers occur okay because those are undefined they lead to complex numbers and complex numbers are not what we are dealing with so far in this course and also just as a generic final statement wherever the functions definition forbids now there are functions that actually forbid certain numbers because of the way these functions are defined for example a later example which we'll talk about in the course is the function natural log of X and natural log of X is undefined by definition for X less than or equal to zero in fact the only way to define it for those numbers is to move on into the complex numbers so there are functions that are actually by their very definition undefined for certain possible numbers but usually the kind of things you will see is you want to avoid division by zero or even roots of negative numbers so let me give you a couple of examples to show you how this kind of works in practice an example find the domain is the question and the function here given again in that standard form G of X is 3 x over x squared minus 4 all right the solution is to find the domain what I want to avoid are the things that were mentioned in my previous list in this one there are no roots involved so I can ignore that also there's nothing mentioned about the definition of the function so the only thing I see that might be a problem is that I have a division and I do not want the bottom here the denominator ever to be 0 so to eliminate division by 0 what do we need see this is a good pattern for you to develop what do you need you need x squared minus 4 not to be 0 well you need to solve this little inequality it's not too hard to solve one thing that will help is if you notice that the left hand side is the difference of two squares and if you remember how to factor that this is X minus 2 times X plus 2 not equal to 0 well the only way these this product can be 0 is if either one of these is 0 and that would be X equal 2 here or X equal minus 2 here so we need that X is not equal to 2 or minus 2 so with that in mind I can write down the domain of this function the domain of G is the set of all the X's and the real numbers such that X is not equal to 2 or minus 2 and there you go I've just found the domain of this function but you see the reasoning I used here I looked at this and said what could go wrong division by 0 could go wrong I do not want the bottom to be 0 so I write that down half the battle here is writing this down once you've got it written down then it's just an equation or in this case an inequality to solve and I don't think you'll have any difficulty doing that let me show you another example one that is similar certainly in style find the domain and remember why we're finding domains after all we're finding domains because they are one of the three parts that make up a function the correspondence or the rule is certainly something we'd like to have the domain probably is the next essential thing here is the function in this case let's use H of X for change square root of 3 minus X the solution to finding the domain well again I look at this and I ask myself what are the possible things that could go wrong there's no division here so I don't have to worry about division by zero but I do have an even root the square root root 2 and so I do not want the inside ever to be neg so to avoid a square root of a negative number which as you know leads us to a complex number we need and again I'm writing this down notice the pattern here we need what we need that 3 minus X is greater than or equal to zero but not less than zero it is either equal to zero which is fine or greater than which is positive well this is easy to solve we could just add X to both sides we have three greater than or equal to X in other words reading it the other way X is less than or equal to three so if that is satisfied then numbers in here remember if X is less than or equal to three the worst that can happen is zero otherwise you're taking a number away from three that's smaller than three which will leave a positive number in those cases this function will be defined so here the domain of H is the set of all X in the real numbers such that X is less than or equal to three and again this is one of those cases where I can rewrite this as an interval including three from minus infinity to 3 and that interval notation that we saw a long time ago is certainly coming in handy now oK we've looked at the domains of a couple of functions let me make some final notes here final notes note number one you might wonder why I'm concentrating on domains and not saying very much about ranges well ranges are often harder to calculate okay so we'll mostly discuss domains because really if you know the domain and you know the correspondence or the machine that the function is then together you can then construct the range so very often ranges are really much harder and not really much use in most of what we do so will mostly discuss domains now these notes the next two notes are technical notes this is a tech note this is regarding your graphing calculator there are two things you need to be aware of one is when you see a key on the graphing calculator that looks like this range it means often window now window is the terminology I will use in the course and most of the modern calculators use window but sometimes you'll find an older calculator that will say range and what they mean by that is the window for the graphing screen so it does not mean range often means window not the range of a function okay so just be aware when you see that it's not the range that you think it is and the final note here is actually a pleasant one some familiar functions appear on keys I won't list every possible case but let me show you some of the ones that I know you'll see you'll see a key that looks like x squared of course this is the function f of x equals x squared you'll see a key that has the square root symbol that is going to be the function f of X equals square root of x you'll see a key that will be either X to the minus 1 perhaps or 1 / X or 1 over X that of course is the function f of X equals 1 over X there are other functions but a lot of them are trig functions or log functions for example or exponential functions which we haven't talked about yet so I wanted to leave you with a couple of notes there on some of the functions as you find on your graphing calculator and when we come back from this we'll practice some of this function notation now the notation of functions is so new that we're going to take a little time now to practice some of it so that you can get some of those older habits about multiplication out of your system so now we'll look at function notation practice and I'll do this largely by example a couple of notes here and there my first example I'll start out with this function let's let f of X equal and I've looked at this function previously 6 minus x over X plus 2 nice simple rational function now let me make a note here off to the side recall this X is in the in the notation of a function ok is a placeholder ok and the other word for that was dummy variable so it might even be better to write this without variables at all as F of box equals 6 minus box over box plus 2 now you'll see why I write it that way I want to do some calculations with this so let me go ahead and do some calculations I may have to bring that picture back let us now ask the question what is f of 1 now remember this is not multiplication this is the F function remember the F function f right F what the parentheses all goes together F of 1 well what was that function f of box equals 6 minus box over box plus 2 now I'm filling the box with 1 so I will have 6 minus 1 over 1 plus 2 okay and of course when you do these it's automatic to simplify them you should make that a habit so this becomes 5 over 3 and we'll just stop there so f at 1 is 5/3 the domain element is 1 the range element is 5/3 let's compute another 1/2 of minus 1/2 again here's the function f a box is 6 minus box over box plus 2 the box is now filled this box right here is now filled by minus 1/2 now when you deal with negatives be sure and use parentheses to be careful so 6 minus minus 1/2 notice I'm using parentheses to be careful and on the bottom minus 1/2 again parentheses plus 2 and of course you can simplify that and whichever way makes you happy this is 6 plus 1/2 on the top six can be thought of as 12 halves so that's 13 halves on the top the bottom is minus 1/2 and 2 2 can be thought of as 4 halves minus 1/2 is 3 halves and then if you remember your division of rationals that we did much earlier in the course 13 halves over 3 halves leaves you with 13 over 3 so f of minus 1/2 the function from the previous page equals 13 thirds again the domain element is minus 1/2 the range element is 13 thirds careful use of parentheses here and here please note that and do it in practice yourself now let's look at another one that's a little different F of the quantity a plus h so the quantity in here has gotten more complicated it's not a single number like these were in fact it's two numbers added together in letter form well if I remember the function let me bring it back again if a box is 6 minus box over box plus 2 I am filling the box with a more complicated object but it doesn't matter that object will go there there and there so f of a plus h will be 6 minus the quantity a plus h again being careful with my parentheses over a plus h plus 2 now in this case I there's no more simplification I can do there's nothing that's going to cancel so I'm going to leave it just the way it is finally let me do one more with this same function just to show you some of the pitfalls here F of 1 over X minus 1 what will that be again here's the function I'm replacing the box by 1 over X minus 1 in all three places so I will now have a fraction over here that will be 6 minus 1 over X minus 1 and then 1 over X minus 1 down here plus 2 and again you can simplify this in any way that makes you feel good I'm just going to get a common denominator on the top and the bottom X minus 1 is common to the top so I will have a fraction on the top with X minus 1 in the denominator I will have 6 times X minus 1 minus 1 and on the bottom a similar fraction this is not the only way to simplify this this is just one way for practice 1 plus 2 times X minus 1 of course when you flip and multiply the X minus ones will cancel each other and then I will have multiplying out the top here 6x then I have minus 6 minus 1 is minus 7 all over 1 plus 2x put the 2x first then I have 1 minus 2 so I have minus 1 okay the domain element the range element and actually this was the range element I could have stopped there from here on this is simplification always wise to do for the sake of argument but remember the answer is really done at this point all right let's look at another function so we can get this pattern here example suppose I have a function capital G of x equals 2x squared minus 3x now in order not to confuse myself I'm going to rewrite it as G of box equals 2 times Box squared minus 3 times box now the structure of the function here is made clear whatever goes in the box here goes in the box on the other side and then you can do a series of examples I'm going to want to make a point here so let me do G of X plus one well G of X plus one remember that's what's going in the box so I'm going to have two quantity X plus one squared because X plus one goes in that box also minus three times X plus one again it goes in the final box two and then just because I want to compare this to something I'm about to do I'm going to multiply this out and I'm going to leave out a step so I'll put dot dot dot here if you multiply this out and I know you can do that you will end up with 2x squared plus X minus one okay I'll box that in so you can see it now let me do something else this was G of the quantity X plus one X plus one went in each box let me now compute G of X plus G of one now some of you may suspect that those are the same let's see in this case what happens G of X is easy I just copy it from up here and I'll put it in parenthesis just to be careful or safe 2x squared minus 3x just copied that plus G of 1 now G of boxes that put one in there I'll have 2 times 1 squared which is 2 minus 3 times 1 which is just minus 3 so when I simplify this I have 2x squared minus 3x minus 1 please note that these two are not equal what's different the X and the minus 3x are what's different but they're not equal which means that G of X plus 1 and G of X plus G of 1 are not the same now that should lead you to a general conclusion let's write it down in general do not draw conclusions about these dysfunctional notation until you have actually proven it in general if you take f of X plus h that's not going to be equal to f of X plus F of H in general the functional notation does not split up like that now the reason that you might think that is because you're used to multiplication and you're in multiplication you multiply F times h plus F F times X plus F times H and that would seem reasonable but in functional notation that's not the same thing at all okay not the same as let's go ahead and write it out say not the same as taking four times X plus h because four times X plus h is indeed four times X plus four times H but this is not that this is not multiplication this is f of box remember the big picture half a box these all go together this is not multiplication all right let's do another example of something you can do with this functional notation here's here's something that might be phrased this way if f of X is given to you as 2x cubed plus ax squared plus 4x minus 5 and you're told that f at 2 or F of 2 is 5 the question is what is a what is that letter right there well you have enough information to do this and the solution is most elegantly put together this way you have F of 2 equal 5 write it the other way 5 equals F of 2 now you have the function here and although I didn't write it with boxes this time you know that X is just a placeholder put 2 in wherever X is using this format I then will have 2 times 2 cubed plus a times 2 squared plus 4 times 2 minus 5 well we can simplify that 2 cubed is 8 so this is 16 plus 2 squared is 4 4 times a this is 8 minus 5 so that's plus 3 so this is 19 plus 4ei I'll put a period there remember this format I said we'd be using throughout the course this is one long equation so this 5 on the left is now equal to 19 plus 4ei let's go ahead and write that down 5 is 19 plus 4ei now we can solve for a move the 19 over we have minus 14 is equal to 4a and then reversing it and dividing by 4 a is equal to minus 14 over 4 or because you can't help yourself it's minus 7 over 2 so this is just another example of using this functional notation in a particular situation how about a definition that will allow us to use it even more definition I'm going to define something called the difference quotient the difference quotient now falls right with the world it ought to involve both a difference and a quotient the difference quotient of a function say just for the argument here called it f at a point we'll call it a here or at sorry at a number don't want to call it a we'll call it C but in quotes so the difference quotient of a function f at a number C as C is in the domain of F is this is what I mean by the difference quotient f of X minus f of C over X minus C now you know that Auto ring a bell if you think back to when we talked about lines in the previous unit we had difference of Y values over a difference of X values to get a slope this looks just like that and in fact this is the slope of a particular line and of course in order for this to make sense comma there's one thing you need to take into account you don't want the bottom to be 0 so you make sure that the X that you choose is not equal to C so that's the difference quotient it's introduced because it's easy it's algebraic and it also leads eventually into calculus but for now let's just go ahead and calculate one with a different given function suppose f of x is given to be 2x squared minus X plus 1 and C is given to be to find the difference quotient and it was called the difference quotient well because there were two differences and the quotient solution well I'm going to need F of C so let's calculate f of C F of C here is f of 2 because C is equal to 2 and here's F again written without boxes but by now you should be catching on that this is a placeholder as are all these other X's so the place now needs to be filled with two so two times two squared minus two plus one well that's easy enough this is 2 squared is 4 times 2 is 8 8 minus 2 is 6 plus 1 is 7 hence we can go ahead and write down the difference quotient f of X minus F of 2 over X minus 2 is f of X now let's go ahead and substitute in 2x squared minus X plus 1 parentheses always keep parentheses until you actually do the operation in this case the subtraction minus 7 because we that's what we figured F of 2 was over X minus 2 and I'm getting a little cramped here I'll move to the next page this will continue to become two x squared minus X minus 6 over X minus 2 and really that is the difference quotient you could stop there but since we've had practice in factoring let's go ahead and see if we can go any further just for the heck of it the top you might try factoring is 2x times act x times something and then if you pick the right values and it turns out plus 3 and minus 2 do it on the bottom you have an X minus 2 of course now X minus 2 over X minus 2 is 1 wherever X is not equal to 2 but the difference quotient remember it's not here but in the original remember we said that X was not equal to 2 so there was no division by 0 well if X is not equal to 2 then these numbers are nonzero this over this is 1 and so you can simplify this to 2 X plus 3 but I'll put that in brackets because that is a good skill to have but it is not essential this is after all the answer okay we've had a little bit of practice now with the notation of functions there's one more thing we want to look at before we start graphing these people these functions in earnest we have to look at how to graph them so the next topic will be visualizing functions and that will be in a moment you you
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Channel: UMKC
Views: 220,965
Rating: 4.8598948 out of 5
Keywords: Functions, table, machine, unique output, Language, Notation, Input set, Output set, Domain, Range, implicitly, explicitly, Set-Builder notation, the difference quotient.
Id: PTf1-bOFPBg
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Length: 84min 53sec (5093 seconds)
Published: Mon May 04 2009
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