06 - What is a Function in Math? (Learn Function Definition, Domain & Range in Algebra)

hello welcome back to algebra in this lesson we're gonna cover the concept of what is a function now in the last section we talked about relations and functions and algebra and how they're very similar to one another and how a function is different from a relation now we're going to take and turn our attention specifically to the concept of a function because this I've mentioned before functions are basically what you're going to use throughout algebra throughout geometry throughout calculus physics chemistry engineering pretty much any type of advanced learning that you're gonna get into and learning is going to involve the concept of a function so in this lesson what I want to do is is go a little bit deeper into what a function is illustrate it for you and give you some examples of very common functions that we will see as we get into more of our advanced study of algebra so at the end of this you should have an idea of what a function really is and examples of some common functions that we'll be using as we go along forward into the class now I could write the definition down on the board but honestly it's very simple to understand it so I'm just gonna read it to you and then I'll draw a picture to illustrate it a function in math or in algebra is a mathematical I'm gonna call it a machine but you can think of it as a mathematical object right or a construct that takes X values as the input to the machine right and it outputs Y values or we say f of X functions of X we call them for now Y values well that's what we've been calling them up until this point in a one-to-one correspondence so what it means is you have a black box here it's something that we call a function the input to the machine is values that we call X and it performs some functional operation and output the output on the other side is another number called Y what we call in a one-to-one correspondence that means that there's only one value of Y for every input value of X and we talked about this a lot in the last section because relations in general are different they can have more than one value of y for each value of x but for functions for every input you put in you only get one output that's what the basic the basic difference is so if we wanted to draw it and we do want to draw it I like drawing things especially with this because it makes it very easy to understand so here we have a black box we call it a machine right but what we're really saying is it's some kind of calculation in here and inside of this we're representing as the function f of X right now this notation does not mean f x x so you kind of have to kind of reprogram yourself a little bit because it looks like f is x x it doesn't mean that it means F is some function it's a function of X so we put it in parentheses to show you that you're taking values of X and you're sticking them into the function now these values of X where do they come from well you feed those values in from the outside and these are the X values so you see you have numbers you're putting in the X values and then you operate on it on this function of X that's why it's parentheses X and the output comes out of the other side so that's why we say this is a machine kind of it taints these numbers and it does something to them and then output the output comes out the other end and so also say that this is the output values and the outputs that come out are called f of X but up until now we've basically been using functions a lot even even though you haven't realized that we've been talking about lines and lines or functions just like most objects in math that we're gonna learn about are going to be functions well you've already used one it's called a line that's a function of X remember we stick the x value in and we get the Y value out so here's where we kind of have to kind of take a step back and talk candidly with you a little bit right so a lot of students get confused when we talk about functions because we've been learning about lines we learn y equals MX plus B right we take the value of x we calculate the value of y and then you get to this idea of a function which is kind of the same thing you're taking the value it's going in you calculate something and then out comes the value out you see what we were learning this concept of functions now it's exactly what you've already been doing it's no real different it's just a relabeling before we said that y is equal to MX plus B here we're just saying instead of calling it Y we're gonna say the output value is called f of X this is the way you read f of X it's a function of X so everything that you've learned up until this point about line specifically applies to functions you take input values you count laid out put values it's just in the past we always called those output values why here we're calling them f of X so then you might say why are we renaming and it doesn't make a lot of sense well it's because as you get more and more advanced math there's a whole branch of mathematics called function theory that that you kind of have to crawl before you walk so when we first started we talked about Y is equal to MX plus B or whatever function that we have we don't really talk about functions in the beginning because we're just getting used to the idea of plotting things on graphs and soap and so on and so forth but as soon as you have that idea of the line under your belt and you basically have already been dealing with your very first functions which are called linear functions functions of letter lines now we can take the training wheels off and say hey what you've been doing in the past is really a function and so what it is it's a mathematical machine inside the machine before we've learned it's MX plus B that's the function that you've been dealing with so far but this function can be anything it can be all kinds of things that we'll talk about as we go more and more through math and we'll learn several of them here but anyway you take inputs you calculate things and you get outputs all right so let's do a couple of quick examples just to illustrate so here's an example all right here's an example function f of X is equal to 2x plus 3 now you should recognize this because if I cover up the f of X part and if I just put y equals 2x plus 3 what would it look like you should recognize that this is a line right it has a y-intercept of 3 and it has a slope of 2 and we know how to calculate that we can either plot the y-intercept and then do rise over run and draw the line we can do that but if it were y is equal to 2x plus 3 you also know that you can write a table you can stick values of X into into this place and calculate the corresponding Y values now we're just relabeling everything and we're saying we're not calling it y is equal to this anymore we're saying f of X means this is a function of X and this is what the function is equal to now this specific function is called a linear function so now you're kind of getting kind of grown up a little bit and you're studying and you start using fancy terms like linear function linear just means line function means its function means it takes an input and you get a single output out there's a one-to-one correspondence for every input you get one input out only one that's what a function is and of course we talked about the vertical line test before because every time you draw that vertical line you only intersects in one spot you only have one output for every input that's what a function is so this is an example of a linear function and if you wanted to graph this it would be basically exactly what we did a long time ago when we started talking about lines we can take the input values and we can list them over here and we can put it a table with the output values so we put the input values of X into the machine and we take the output values as we calculate them on the output side and then we're gonna have XY points although we don't call them X&Y anymore we call them X and f of X but you plot them as X and Y's we always have so we can for instance take negative two is an input negative 1 0 1 and 2 and of course we can go on from negative infinity to positive infinity calculating values until we're you know blue in the face so how do you calculate this you take the negative 2 into the function which means you put it into the value of x which means what you have here is the way you write it is f of negative 2 means you're taking negative 2 and you're sticking it into the function this does not mean F times 2 or an F times negative 2 it doesn't mean this is multiplied this means that we take the negative 2 we stick it into the function which means we substitute it into the position where X is so we have 2 times negative 2 plus 3 so what do we have here this is negative 4 plus 3 what do we have here negative 1 right that's what we get for the first value and then for the next value we can say all right let's take F and we'll stick negative 1 in so it's F of negative 1 that's how you say it out loud so it'll be 2 times sticking negative 1 in here plus 3 this is gonna be negative 2 plus 3 and what do you get here you get 1 for the second value all right and then the third value we're plugging in a value of zero so it's f evaluated at 0 or F of 0 is equal to 2 times 0 plus 3 this is going to give you 0 plus 3 so we can just write 3 down that's the easiest way to write that and then we can now evaluate at X is equal to 1 so we say X f of 1 is 2 times 1 plus 3 so here you have 2 plus 3 is 5 and the last one F evaluated at 2 make sure you read it as F of 2 2 times 2 plus 3 this is 4 plus 3 is 7 so I can kind of shortcut a little bit there so you have negative 1 1 3 5 & 7 what do you think this is gonna look like when you plot it I mean you already know it's gonna be a line but just to for completeness let's go ahead and quickly draw it real quick we're not gonna use the big graph paper because we've done this so many times so we have X but instead of writing Y up here which is what we've done before we'll write f of X so you're plotting for the x-values you're plotting the x-values and for the y-values you're just plotting these f of X values so for functions you need to start thinking of x and f of X what the function has done it's operated on the values of X it spits out an output value that you call f of X so you plot x verses f of X so now we have the values of x here we'll just go ahead and start I guess I should have drawn it in the negative direction as well right because what we're really plotting is let's go negative 2 negative 1 this is 0 this is 1 and this is 2 and it looks like our Y values go up 2 let's just go up to 1 2 3 4 5 6 7 this is 7 this is 5 this is 3 this is 1 1 2 3 4 5 6 7 and we have a little bit of a negative value as well so with a negative 1 so what we're gonna do is plot it negative 2 comma negative 1 so negative 2 is here comma negative 1 it's a little hard to write but it's basically right around here and then what we have next is negative 1 comma 1 which means negative 1 for X positive 1 for y which is right around there then we have 0 comma 3 so 0 comma 3 1 2 3 is about right around there and then we have 1 comma 5 1 comma 5 is around there and then 2 comma 7 2 comma 7 is around there now I know it's not perfect and it's I'm actually really close to the board so I can't really read them too easily but it's a straight line you can more or less see it's a straight line so I'm just proving to you even though we've done in a million times before with lines that when you have something that looks like 2x plus 3 for your function that's what we call a linear function so you need to start using and understanding the terminology of algebra as much as this much of what I'm trying to teach you here is the terminology in addition to the math right so when you see something like you have a linear function it looks like this it sounds very complicated all it means is it's a line right so if you ever have something that asks you once you learned about functions in your exam or something and it asks you you know here are two points you know write me an equation of the function here well it's exactly the same as writing the equation of the line that you've been doing over and over again first you find the slope between those points and then you write MX plus B or you can use the point-slope form Y minus y1 equals M times X minus x1 and you basically do it exactly the same way it's just that instead of saying Y is equal to this you say f of X is equal to this so we're doing mostly terminology here but from pretty much here on out we're gonna be talking about functions functions of X right so this is called a linear function I want to go through a couple of additionals just to give you some examples of what other functions might look like let's say that we're going to take input values of X and we're going to calculate what the function looks like if it looks like f of X is equal to x squared now the first thing is you should notice that this is not a line at all I mean the equation of a line is MX plus B number times X plus number right but this looks nothing like that because you have an exponent here so right away you know that all lines should look like MX plus you should have that in your mind this doesn't look like that so it's not gonna look like a lie right so we're gonna go ahead and show you how that looks right now though just to keep it on even footing we'll choose the same exact points negative 2 negative 1 0 1 & 2 as values and we'll stick them in here so we'll take and operate on the first one F of negative 2 is negative 2 squared make sure you stick the entire negative quantity into B squared negative 2 times negative 2 is positive 4 here we'll have a function evaluated at negative 1 stick the negative 1 in there negative 1 times negative 1 is positive 1 we'll take the 0 in there we'll evaluate it at 0 0 squared 0 times 0 is of course 0 switch colors just to break it up a little bit we'll take F evaluated at 1 is going to be 1 squared 1 times 1 is 1 and then in 2 you can see the pattern here it's just gonna be 2 squared which is 4 now notice the interesting thing about this x squared function here even for the negative values of X we ended up getting positive values on the output f of X but the positive values of course we always get positive values so the interesting thing is because you have a function that's squared no matter what the input is if it's positive or negative you're still gonna get a positive answer so let's take a look and see what this graph looks like just to compare it here so we will go and look over here and see what this looks like so let's say we have here's negative 1 here's negative 2 here's 0 of course and here's one here's 2 and let's say we have 1 2 3 4 let's call this vibe right now let's plot this guy and take a look at what we have negative 2 comma 4 means a negative 2 way up here at 4 that's what it looks like at the beginning then negative 1 comma 1 negative 1 comma 1 is way down here then we have 0 comma 0 that's right at the origin then we have 1 comma 1 1 over here 1 up is right there and then 2 comma 4 is up here now it looks like you could just draw a little light lip sorry little line segments to connect them like a little line segment here little different line segments here different line segments here a different line segment here and it does look like that at first glance but if I had put more points in here like if I put negative one and a half if I put negative 0.5 if I put point five if I put one and a half or if I put like a hundred points in there in between all of these numbers to get a lot of granularity and exactly what it looks like what you would actually find is the function would trace a path more like this it would be a curved kind of deal that kind of goes and it just kisses the axis there and then it goes back up like this it gets very steep at the ends here this guy is not a linear it's not a linear function that's called a quadratic function we're gonna talk a whole lot about quadratic functions later on so don't worry about it I'm just giving you the the name of it so that you can kind of get used to it roll it off your tongue a little bit but the idea of a quadratic function just means that you have X with a square power any function with X that has a square power we call it a quadratic function and they're pretty much always gonna look something like this sometimes they might go upside-down but they're always gonna be a smiley face or a frowny face like this kind of doing this now notice he gets very steep up here and over here because everything's squared so here when I starting if I would go to the next the next one it would be 3 comma 9 we have 3 times 3 is 9 or 4 comma 16 or 5 comma 25 you see what's going on here 6 comma 36 because you're squaring it so as you get farther in X it's getting really really steep so this thing gets really really steep as you get just past and go into the positive territory or in the negative territory here as well alright so that's a quadratic function now what I want to do is I've Illustrated by plotting points to show you hey this is called a linear function you've already talked about it it's a line here we have I have we haven't learned much about it but it's called a quadratic function by the way if you want to see that written down it looks like this quadratic function right but the point of this lesson isn't really to learn too much about this or too much about this the point of this is just for you to get the idea of a function in your mind and have some practical examples of what functions look like so what I want to do before we close the lesson is I want to illustrate what a few more functions look like kind of a tour of what you might see some functions that might look like in algebra and beyond all right so here's some functions that we can talk about we can say we already talked about f of X is equal to x squared we just drew it here but just for completeness I will go ahead and put it in its place a quadratic function it looks like this he starts up high it kisses the x axis and then goes back up high again and you saw that by us plotting our points just like we did a minute ago of course we already talked about the linear function which is just a line now let's take a look at what this function over here looks like f of x equals x to the third power so it's not x squared it's X to the third power this is called a cubic function and if you were to plot this of course it's freehand so forgive me a little bit it's actually gonna look pretty interesting it's gonna start down low it's gonna go kiss the axis and go through it and then it's gonna go up this direction right now what I want to do is I want to get all of these on the board and then once they get them all on the board I'm gonna go back through them and kind of explain why they look the way that they do but first I just want you to kind of get an idea of hey here's what a quadratic function of x squared type function looks like here's what an X cubed function looks like let's get through them all and then we'll go and talk about them a little bit more let's say we have f of X is equal to X to the fourth power because you can have these quartic functions we call them we got this guy right here and so we can draw this guy now it's interesting as well it's gonna be really really really yeah let me do it like this really really really steep and they're really really really flat and then really really really steep again notice it kind of looks a little bit like a quadratic it's just much steeper and then at the bottom it gets really really flat and we'll talk about that later I haven't drawn any numbers I don't want to get into the details here I want to show you kind of the overview of what these look like another one I want to share with you is important f of X is going to be equal to 1 over X alright talk about this later but I want to just draw it for you right here's X and there's f of X now this one's interesting it actually split into two parts it goes really a PI like this and then curves down and then on the negative side it goes like this some kind of little draw a little Haros here if you like I'll explain why it looks like this in just a minute but that's what the 1 over X looks like this is a hyperbola and then we will talk about one last one right and the last one I want to talk about is f of X is equal to the absolute value of x right what do you think that's gonna look like absolute value of x is a kind of a triangle shape thing it's a straight line sorry I put a little curve in that I didn't intend to let me try to draw that a little bit better it's a straight line it's best I can do anyway and then it's a straight line this way mirror image reverse and it has a hard like a point right in the center there a little cusp right there now one little bonus one I'll drive will do this on the last page because I really want you to understand the idea behind a lot of this stuff and where we're going with this right here's X here's a function that you won't learn until much much later in trigonometry at the end of algebra 2 and also in trigonometry it's called the sine function sine of X this is kind of a bonus you're not going to use this too much now but I want to show you that these functions can look pretty interesting this function starts here and it goes up and it goes down goes up goes down and it goes like this forever so in the reverse direction it goes like this and so on it's a wave basically so just a little bit of a fun fact for you when you get into more advanced studies and you talk about radio waves any kind of wave that you might have like study in nature it looks like this which is a sine function notice it passes the vertical line test just like any other function you can only cut it in one place but it goes up and down and up and down so this is what a radio wave would look like traveling to userspace well let's come down to earth a little bit and talk a little bit about these functions this one makes sense because we already plotted it and so it's doing this because we're squaring everything that's our everything getting very very large this one is looking like this because it's cubed so when you look at the positive values of X it just it's just multiplying by itself three times so if X is 2 then it's 2 times 2 times 2 if X is 3 then it's 3 times 3 times 3 if X is 4 then it's 4 times 4 times 4 so it gets really really really big really really fast but on the negative side notice it curves down like this because if you have negative 1 then it'll be negative 1 times negative 1 times negative 1 so negative times negative times negative will give you a negative value it'll still get really big in the negative direction but it'll be negative because it'll have again on the negative side negative 4 times negative 4 times negative 4 will give you negative large number for instance so it's gonna get really really big in the negative sense and that's why it doesn't quite mirror this it kind of starts in the negative sense and then it goes back up in the positive sense now this one is to the 4th power that means it's getting even steeper than this because you're multiplying it times 4 so if X was 5 it would be 5 times 5 times 5 times 5 so it gets really really really steep fast but in this region it bottoms out so flatly because it between negative 1 and 1 when you have decimals there if it let's say X was 0.5 then it would be 0.5 times 0.5 times 0.5 times 0.5 so in the decimal regions in there you're multiplying by a decimal over and over again so it cuts it down and gets it really close to 0 like a bathtub down here that's why it gets so flat here this one does it as well but not quite as much as this one now this one's called the hyperbola because you're dividing by X so when X gets really really big like 1 over let's say X is 100 1 over 100 it gets really close to 0 but as as X gets really really close to 0 over here like let's say let's put X is equal to 0.5 then it would be 1 over 0.5 let's say X is 0.1 really really close to 0 let's say X is point oh-oh-oh-oh-oh-oh 1 then you say 1 divided by 0.001 which gives you a really big number so you divide by the small numbers and gives you a big number you divide by the big numbers gives you a small number exactly the same thing happens over here but with negative numbers so that's why when you're dividing by the negative of X it does the same thing but over on the negative side for Y and then the final ones the absolute value because over here it makes sense what's the absolute value of two it's just going to be two what's the absolute value of three it's going to be three what's the absolute value of 17 is gonna be 17 but over here what's the absolute value of negative 5 4 if X is negative 5 it'll be positive 5 and so on so any negative value you put into this function gives you a positive value out so the point of this was not for you to memorize these functions the point of this was just to show you just kind of a potpourri of what some some functions look like so what we're gonna do is we'll be talking a whole lot more about these linear functions as we go through the class we've already done that lot then we'll be talking a lot about quadratic functions because quadratic functions with with the parabolas like this they make up quite a bit of science and math right so we'll be studying that a lot but along the way we will get into cubic functions and you may have some other hyperbolas and things like this and then when you get on into advanced math you'll have crazier functions that look like waves right but the point is for you not to memorize them the point is for you to know what functions are every one of these functions passes the vertical line test slice any of these functions vertically and they only get cut in one place even this one it only gets cut in one place so their function so that brings me back full circle a function is a mathematical machine that takes X values for input and outputs Y values for output in a one-to-one correspondence meaning for every value of the input X you only get one value of the output and when you plot that then if we say that it passes the vertical line test and that's a function so make sure you understand this follow me on through the rest of the lessons we'll be talking about functions for the rest of our time together so I want to introduce it here and then we'll get to the following lessons where we talk about different types of functions and how to solve equations with functions and do all kinds of things all of which are really important in all branches of science math and engineering

Channel: Math and Science

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Keywords: what is a function in math, what is a function, whats a function, math, algebra, algebraic functions, what is a function in algebra, domain of a function, domain and range, range of a function, domain and range of a function, graphing functions, function notation, function math, mathematics, algebra for beginners, algebra tutor, linear function, quadratic function, linear equations, algebra help, help with functions algebra, function definition, definition of function

Id: GY6Q2f2kvY0

Length: 26min 17sec (1577 seconds)

Published: Wed Dec 19 2018