I welcome to another video in this video we're gonna continue talking about functions but not really just what they are how to evaluate them and we're gonna go past just the whole plug in a number we're going to see how to evaluate them for different expressions and sometimes we call that composition of functions so this is exactly what we're gonna be talking about later on in the class we say let's compose some functions but right now we're gonna look at it under the lens of composition so the main ideas I'm trying to get across to you is that evaluating a function takes a function and replaces the variable the independent variable with something else whether it's a number or another expression it just replaces it so we're gonna we're gonna what we're gonna do is we're going to take the variable open it up with a blank space we'll talk about that will plug in something into that and then we'll simplify also what I want to look at as we're coming through this is when are there times when there are some numbers that we can't plug in and you've chosen it and say I want to plug this in and then the function is less you because it deals with either an imaginary number or some sort of a discontinuity something that says I can't even get out of something out of here it's undefined so we're gonna look at that so I'm trying to open your eyes to how to evaluate how to do the same exact thing we're evaluating numbers or other expressions variables in them how to simplify those things and then also a little bit about domain we're gonna start exploring how the domain works the last example is really important the last example because this is a precalculus class write a college eldritch leads right into it anyway we're gonna look at something called a difference quotient and it's a very very important concept in the first part of the calculus one when we talk about the slope of a curve at a point the difference quotient when they take a limit of that it's what gives us that idea it's called derivative and you deal with it exclusively in calculus one almost exclusive they tell you this one so let's start looking at evaluation of functions before we go on to just plug in things in I want to tell you and show you why we have two different notations that represent the same thing so for instance y equals negative 7x plus 5 or f of x equals negative 7x plus 5 that's the same y equals 3 halves X minus 1 G of X equals three halves X minus one these two functions and likewise these two functions and they are functions notice no square root with weird plus and minuses know why swear nothing to solve for these are all functions these two functions and these two functions respectively represent the same exact thing the reason why we prefer to have function notation is is twofold number one it's so that we can distinguish it I think I made a joke about Jim in last video like if everyone's named Jim you say hey Jam if every function was named y it'd be very hard for me to distinguish between them now ignoring these that I use the same exact letters let's say that just look at this and say hey check out the function f f of X there's there's no ambiguity as what I'm talking about I am talking about that function or say look at the function H of X we're talking about that function or look at the function G of X we're talking about this function it names them by a name f or G or agent not a great name some one letter but it names them by a function and not only that it's more descriptive of what you're dealing with when you see a function letter - followed by a variable it's telling you what the independent variable is that you're looking at so this is hey there's gonna be a function I'm gonna identify it by this letter F and more than that it's gonna be dealing with a variable X so this is the variable I'm gonna look for the second reason and equally as important is that it's more expressive in what it lets you do when you evaluate so if I tell you plug in two you could all plug in two and say or let X equal two or evaluate the function for two so let's plug in two can i plug into here oh yeah clearly I can we get negative seven times two plus five and we would get y equals let's see negative seven seven or six negative 14 negative 14 plus five looks like native nine let me go great white was native night that's perfectly fine we plugged in number two we got on an answer or output negative nine we could plot that point to negative nine it's it's great but this function notation gives us a bit more you see and here's the whole thing we're gonna be doing so please watch us it's not trivial when I give you this notation F of two that f of two tells you exactly what to do I don't even need that phrase it says I want you to look at the function f got it I want you to look at what the variable is Oh X's okay cool and that's why there's X's over here and I want you to replace this X with 2 notice the notation the notation literally tells you what to do says in the function f replace your independent variable two here's the function f with an independent variable X it says I want you to replace that variable with two wherever you see it so it's more descriptive and what it wants us to do so this says hey I want you to replace the variable with 2 that's negative 7 times 2 plus 5 no problem it's the same exact math guys the same things we've done but it's describing it better and then it says hey when you plugged in the two you got out 9 it says here's your function I want to replace the variable with two no problem I replace the variable with 2 it showed me not only when I get out of it but also what I plugged in we sort of lose that over here so sorry boom you sort of lose it over here don't we so unless you read back and knew what you plugged in and you were just given this information you would not know what you plugged in you would not know your input this notation gives us both the input and it gives us the output and if you read it left to right it will give you your or impair as well so again the reason why we use this because we can talk about different folks on the same page and understand that the each have a name and they're not all just Y or Jim and also it's much more descriptive it tells you what your independent variable is it tells you what you can you they want you to plug in for it and then it tells you descriptively what you got out so this says I had an input of 2 I had an output of negative 9 and it's given us an ordered pair right there so those are reasons why we use function notation and because of those reasons it allows us to manipulate our evaluation a lot nicer it gives us a manner in which to do this so here's what we're going to do for the rest of these again what we're exploring is are there some numbers that don't work so some domain issues can we plug in things other than just numbers can be evaluate for other functions that's composition and yeah we can't and what's the best way to do that so we're going to take a look at that so it's let's consider H of X so say hey I got a function whose name is H I have an independent variable X and that's all we see all these X's over here that's the independent variable that we're going to evaluate we're gonna replace that and just like we replace this X with 2 we want to replace this X with whatever I tell you to do or whatever the the the book tells you to do or whatever whoever it's giving you this stuff is telling you what's do there's just me so this says I want you to look at this function and replace the X with negative 1 for the first problem or negative X for the second problem or X plus 1 for the third problem before we go any further it's really common for students to try to do all of this in their head I get it we don't spend a lot of time on it but man that leads to just a multitude of problems so I'm going to give you a really really clear way on how to write this verse two-step process how to write the function first so that you evaluate it properly please to follow this I promise it'll work better than just do it on your head what are we essentially gonna do yeah we're gonna put negative 1 here and here yeah we're gonna put negative X here and your X plus 1 here and here but the way that we do this in structure in your head I can help if we do the phone relay so whatever I know what I know and functions it that is that the variable is the only replaceable thing that we have and so if we want to replace it well maybe we start by giving it a blank space like like Taylor Swift right she's got a lot of blank spaces and I'll write your name so what we're gonna do is we're gonna replace that variable with just a blank space that allows us to plug something into u-men you know what that's the way math works anyway these variables are plug in a bull so if we take this negative 2x squared plus X minus 1 they say the negative 2 is not something I can ever change but my variable the thing that can vary is X and if I take that X and I say hey that's the only thing in this function that I can actually replace let's just let's keep it blank for now in your head right now I need this and this to be exactly the same thing for instance if you were to plug in the value 1 you would put it here and here if you were to plug in the value 1 you put in here and here and you would get exactly the same thing out does that make sense to you like if the variable is the one thing we plug in for that I can replace it for this plug in spot this blank space that we can write anything we want that's important now why do we use parentheses every time well parentheses maintain operations so like they maintain multiplication and maintain exponents they also maintain signs so we can plug in negative they maintain that for us so the first step when we're evaluating look at your function how do you even care what's here right now look at your function and replace your variables big blank space give it some sort of parent that it looks fresh replace it with a blank space so negative 2 got it X no blank spaces squared yes this stands for x squared and if we wanted to write all math as blank spaces instead of variables you could it just get really awkward so we kind of use a variable to represent a blank space that's what they mean they can take any value a blank space can take any value same stuff plus plus X now let's give that a blank spaces I'm going to plug in something for that minus 1 minus 1 that's a constant it can't change then what we do once we replace the variable with a blank space we're gonna look at what they want us to put in there it says right here says I want you to replace X so in your blank space put negative one and that's exactly what we're going to do so we put a negative one everywhere where we have that blank space and then we evaluate it just be careful okay just be careful when you're evaluating don't go so fast you're making mistakes follow the order of operations for instance we don't want to multiply here we want to exponent here so negative 1 squared is C positive 1 positive 1 times negative 2 is negative 2 plus negative 1 minus 1 negative 2 plus negative 1 so we did negative 1 squared is 1 1 times a 2 is negative 2 plus negative 1 minus 1 is negative 4 what I would like you to do is write that as an ordered pair we had an in point when you were given a real number input we're given negative 1 we got up negative 4 and that would be a point on that graph that's an upside-down parabola and that's gonna be one point on that graph so when we're evaluating you're just looking at you say okay I'm going to replace the variable with something it's telling me right here but the appropriate way to go through it is don't do it in your head it might be OK for negative 1 but when we get these expressions especially stuff like this it's really hard to get all of that down all at one time and not make mistakes on so are you seeing this as a valid method like just replace your variable with with a Taylor Swift blank space and then write the appropriate name in there in this case it was negative 1 why I want you to learn that is because it works for everything like ever so if I want to evaluate H for negative X I'm not even gonna look at the negative x don't let that confuse you don't let that ruin your your math day ok all this says is all this says is take your function H ok ignore the ignore the negative X for instance if I said I don't want you to plug in anything I mostly leave a blank completely blank whoa is a blank space and I could take this a safe I want negative two and I want an open spot for necks here's Mayo two here's an open spot for X plus another open spot for X minus one and all it's having to do is say now I'm gonna put whatever's here in here if there's nothing there that would be nothing there that's what the blank space comes from if it's negative X like you wanted then after creating a blank space for your variable we're gonna take whatever's here we're gonna plug it into the variable spot or the blank space in our function and lastly we're going to simplify as much as we can so wouldn't we simplify that again just like we squared negative 1 we would square negative x so negative x squared is negative x times negative x negative x name is a positive x times X is x squared so we get negative 2x squared plus negative x you need to know that when you're adding a negative it's that's the same definition of subtraction so negative 2 times x squared minus X minus 1 and that's exactly what this function evaluated for negative x would be why do we learn that who cares about negative x well in just a while probably 12 beads in just a little while we're going to be talking about even functions and odd functions what are those don't worry about it no but this is how you test for an even or odd function or neither if you plug in negative x and it gives you the same exact thing as your function you have an even function if you plug in negative x and it gives you every sign opposite as your function you have an odd function and it's a mix up of those things you have neither so if you want to look at it right now that's the same sign that's a different sign that's the same set up that's an either that's not even that's not odd that's how you determine it so that's what I want to be able to do that we want to be able to plug in negative x and test for either odd how about X plus 1 let's do that hey you know what you should try right now you should take your phone and when I'd love it if you could replace your variable with a blank space so don't worry about the X plus one don't even I don't care what that is right now whatever it is malignant I replace the variable with it like that's that's how evaluation works I've been with negative one I would have done if it's a zero I did it with negative X we do the same thing with X plus one it's one sort of creature that we're gonna put in there so don't worry about it right now create your blank space so we do negative two blank space squared plus blank space for a variable minus one this now represents exactly the same thing as H of X I just opened up my variable because the variable is the only thing you plug in a thing into anyway now I look back and say well what do you want me to plug in well I want you to replace the variable with X plus one so in that blank space we have X plus one this is going to take a bit more simplification than the previous ones because we're gonna have to distribute and so when we're doing this you know that's a bracelet when we're doing this you would square this first please for the look so don't give me x squared plus one begging you'd have to distribute this first and then you'd have to multiply by two and then we add X plus 1 and the minus one so I'm gonna deal with the X plus one squared first X plus one squared means X plus one times X plus one you cannot just give me x squared plus one it doesn't work this is x squared plus 2x plus one that's what this X plus one squared would do now if we take the native two times that so I've taken X plus one and squared it I figured out this expression I'm now taking that negative two plus X plus one minus one I would distribute this first that's negative 2x squared minus 4x minus or because I'm sorry minus two because we are distributing to all of those three terms plus X minus plus one there's no sign change there's nothing to distribute minus one then we can combine our like terms so that's negative 2x squared minus let's see there's no like terms of that I have a minus 4x and then plus X that's minus 3x I have a minus 2 a plus 1 and a minus 1 those are gone I get a minus 2 this is what this expression would be so H of X plus 1 works out to a different function it's a composition of functions and it would be exactly that that's what we mean by evaluating functions you can do put in numbers you can put in expressions you can put other functions it doesn't matter you can put in whatever mrs. finally face if you want to you put a smiley face in the variable spot it doesn't matter there is just symbols guys so keep that in mind I hope you're seeing the value of that blank space it will allow you to evaluate anything that you want even composition of functions what we're gonna do now is we're gonna move on I told you a little while ago that what I'd like you to see is not only how that that blank space for evaluating is really nice but I also want to start exploring something that we deal with in calculus and some things that would help you see domain issues as these two so I'm gonna move a little bit quicker because the the fundamental idea is the same we're replacing the variable with with a number or another function so let's let's give it a try I'm still gonna be looking at oh oh I haven't said this one true/false this is a non function because it has a square root this is a non function because it has a plus or minus yes that's a little bit so the square roots themselves are fine so we have some notation issue that would systematically give us two outfits for the input that we'd have a problem that right there's a function what I'm asking you to do here is take that function and replace the independent variable in two spots with five and then zero and then to X and then X plus 801 we're gonna learn some things about that let's start with the five so I'm taking my square root that can't change this says don't even worry about these things they're all going to look identical I'm going to replace the variable with some sort of a blank spot a blank space so that it allows me to plug in or add I wait so this says hey x squared minus 3x or something squared some some new value squared minus three times that same new value and it tells you right here what to plug in for that and that's five so when we evaluate f of X for five we've made our blank spot we've then looked at what that was it plugged in there twenty-five square root this group is going to be there five squared is 25 minus 40 is 10 this would actually give us an ordered pair we plugged in five we got out the square root of ten that's a point I hope you guys get the point this is something that's gonna be on the graph of that square root it is going to be there what about zero now we're starting to learn a little bit about what you can and can't do what about zero if we take the square root of x squared minus 3x that's a bunch of that and say I want you to evaluate that for zero so I want you to have some blank spaces something to plug in for so the same exact thing we just didn't notice all of these are gonna look identical it just depends on what you're plugging in keep that in mind I don't care what this function is you do it exactly the same way you're going to replace the variable the blank spot and plug in what I tell you or what the function is telling you here if it's gonna be zero so I'm going to plug in zero and zero and that says away then zero squared 0 minus 3 times 0 that's also true that's a zero is it okay to have the square root of zero is that okay the answer is yes the square root of zero is fine and we can prove that because the square root and the square are inverses in zeros square gives you zero then the square root of 0 is also 0 that's fine zero whereas 0 therefore the squared of 0 is is a valid that's a real number this would give you 0 and so in ordered pair would be 0 0 you plug in 0 you got on 0 that's gonna be actually the starting point of this graph or an end point of the graph that you want to call it that alright moving on I hope you feeling good about it try the next on your own try to take this right the function give yourself a blank spot and plug in that to X and simplify this good work for you to do for us rounder right now so I'm saying I don't really care what this is I don't care I just know that I'm gonna replace the variable with that thing so I'm going to create this open cavity this blank spot for me and I'm going to replace that X or that blank spot with whatever's in this parenthesis so here's 2 parentheses this says this is what you put inside that notice it's also input this news that's convenient so 2x + 2 X and when we simplify that like I think I told you at the start of our the introduction to that this series is that I'm going to try to do several things that I'm teaching you I'm gonna try to remediate when I'm trying to point out common mistakes one of those is that when we square expressions that are multiplied together we need to know that exponents do distribute but only across multiplication and division never addition subtraction that's true for all distribution distribution distribution of one mathematical operator is always distributed across something below it just one level below it so for instance exponents distribute across multiplication division well multiplication division distribute across addition subtraction that's what it works so yes we can distribute that it has to go to both the two and the X and you're done what that's done it's just evaluating this function for 2x or replace the variable 2x and simplified now why that why that this is the start of learning about something called the difference quotient I mentioned a little while ago that the difference quotient in math it looks a little different I'm gonna do one example but how it actually goes how hell goes and what you would do with it when you get to calculus one but this is the start of it so we need be really really proficient at evaluating X plus h in some sort of function now it can look really confusing the first time you see it and you go like that how would I do with that do I just add H at the end no imagine that this was anything else any other number like seven or 2x or whatever or not even there what would you do with it what would you do with your function if I said set this up so that whatever I put here you can put in your function you create a blank space just like what I'm showing you you create this something squared minus three times something you would give it an open door for your variables so that whatever I tell you you can then fill it in that would be a great way to do it so when I tell you if I want you to fill in X plus h you don't have a place for it you do X plus h and you do X plus h and we're good to go now it does look funny because there's more than it looks like there's more than one variable I will talk about in the difference quotient is and what we do with it later but it looks like there's two variables can you still simplify it I hope so I think so we should be able to we should be able to take this and square it X plus h times X plus h let's see that's x squared plus 2x age you can do it x times X is x squared X plus h x plus h bar sorry x times h x times h has two x times h plus h squared this right here is what I'm going to have for here so we just squared that we distributed it we combine some like terms - let's distribute that 3 3 X - 3 that right there is as far as we can go right now there's nothing else to do we're not going to do anything with that right now but it's good practice on being able to evaluate so keep that in mind it's super important going forward that you can replace a piece of your function your variables with another function like X plus h last one here's where we learn a little bit about domain have you tried it yet I hope you have at least thought about it try plugging in one or evaluating this function for one so take it and give it any blank space and - 3 times with makes nice and go alright yeah that's exactly what we've done every other time let's just plug in 1 now then you go 1 square is 1 1 minus 3 is native to is the square root of negative 2 possible it's a trick question is it possible yes is it possible over real numbers no and so how we're defining our domain right now our inputs that give us a real number output since that's not a real number we would say that 1 is a problem for us one's gonna be an issue because I can't plug it in and get a real number a graphical number out because we're talking about graphing functions that's a problem so right now in your head I'm trying to build something I'm trying to build that not every number is possible to plug in to every function we have these domain issues and in general we look for them in two spots one of them is going to be numbers that give you negatives inside square roots not cube roots that's fine but numbers that give you negative inside square roots are going to be problem embers they're outside the domain in general okay what do you want what do you want for square roots well we we have to have the inside of the square root 0 so we talked about this one or more 0 or positive and so we're going to start defining that notice what I'm not talking about please make a note here notices how the number one is a positive number it's not negative but when I plug it in it gives us a radicand the inside of the radical that is negative that becomes negative that's the problem so this is an issue right here so when we look for square roots we want numbers that make the inside positive this is a problem for us so this would be outside of our domain we can't plug in one this would be a non real number it's a magic number but we can't deal with that so I want to start thinking about that right now lastly for goodness sakes you should be doing that on your own you really should try these pause the video see if you can plug in 0 value for 0 or x squared or 5 or this is a little weird we'll talk about that one but at least the first three now I'm going to do it right now but you should be positive ok so G of 0 G of 0 says take this function of G of X replace the independent variable X with 0 so I'm going to give myself a blank space to do that I always use a parenthesis because it makes operations and it maintains a sign I never have to worry about it even here I'm using parenthesis we always do that and it says now that you've replace your variable with a blank space something to plug into I know what to plug in I can plug in a 0 2 times 0 is 0 plus 1 is 1 0 minus 5 is negative 5 so this gives us 1 over negative 5 or negative 1/5 that's perfectly fine you can write an ordered pair 0 comma negative 1/5 now is that in the domain is 0 in the domain is 0 a valid input for this function did it give you a real number output yeah negative 1/5 is fine that's a real number it's not imaginary it's not undefined it's something that we can plug in to get a number out how about x squared same exact idea we can take our function we can replace our variable with a blank space something that we can plug in 2 we now say hey I know one plug in it's telling you right here to replace your independent variable X your blank space with x squared it's really not even much to simplify here all I did was want you replaced that variable with something else and they even told you so when we replace that best to x squared plus 1 over x squared times 5 and you're done there's no simple there's no factoring you can do no canceling out for goodness sakes don't cancel out your x-squared you're done that's all that that's asking you for is to compose two functions to put one inside the variable of another that's it mm-hmm now five we start we go right I know how to do this I know I'm gonna take T PI function and replace my variables with something so I'm gonna I'm gonna give it a blank spot because that's what we do right that so we do we plug things in we replace our variable with that thing so we imagine that we replaced our variable we give a place to put that we plug in five because that's what it's telling us right here it says replace your variable with five and then we simplify and on the numerator we get a little tip so it's ten plus one yeah let it over zero let me go yeah that's that's zero right oh whoa hey God have you seen have you seen this before some of you may know them all of you probably have heard this at least but maybe don't understand why you can't divide by zero so I'm gonna take a moment and teach you this if you don't know it can you do zero divided by four well if we understand that the denominator of a fraction is what you cut something into so like imagine pizza can you cut a pizza into four slices sure you can there's a pizza with four slices can you then take zero of them yeah yeah it's called a diet you just don't eat the pizza it can you take zero out of four slices mm-hmm it's a diet how much pizza have you had you've had no pizza that's possible that makes sense now imagine it next one okay imagine this says you have no pizza no you're doing to have a slice this says there's nothing there and you can't imagine like my kids do like a fake pizza all day long right it's delicious so good no no this is there's no pizza and yet you're literally taking four slices this doesn't even make sense you can even draw a picture of that here's your picture there's nothing there and now take four of that how much have you don't have zero because you can't answer the question this says you have four of that okay that's what that's trying to get says no you don't have zero its dancers non zero because this question can't be answered this says you have nothing there and I need you to take four of that how much do you have you know I can't do it it's not even defined you've defined no pizza so when we get to something like this it's it's not a real number it's I think an imaginary number but it's undefined so when we start talking about the domain what we're looking for are the numbers that we can plug in that give us real numbers and defined numbers out so is five in the domain yes give us an out it's zero in the domain of that function yes it gave us a real number out is one in the domain of that function no it gave us a new real and imaginary number out is zero in the domain of this function sure is five in the domain of this function no because it's not even defined that the answer that outputs not even defined so this would be outside the domain one thing I want you to understand before we go any further what I've tried to show you is that you're going to deal with until we get to logarithms to issues with our domain number one is square roots we really have to make sure that the inside of the square root is always positive and then we plug in numbers that make that happen the second issue is with denominators we can never have an input that makes zero on our denominator those are the two problems that we need to address moving forward those are the two issues that when we talk about domain in the next video are gonna be really like neon sign saying you're gonna have problems here square roots and denominators and I promise you this if there are no square roots and there are no denominators your domain will be all real numbers in tele2 logins so all real numbers there's no square roots as no numbers square roots we're gonna have some problems do not we could have some false denominators we could have some problems there those are the issues we're gonna be looking for notice one more thing really quick is that you can't have zero on the denominator but plugging in zero could be fine notice I plugged it in it's not about Zero's about what you get out of it so that's about it as far as our our domain talking the last thing I think I've raised here is is this don't confuse this problem with affecting your input notice that this says I'm not affecting my input at all here are places a five here at plays an x-squared here a place of the zero I'm not changing my input at all what that means is to take your function and just put a negative in front of it that's all that does now you can move that negative around you can distribute it to a numerator or it's an honor but not both or you can leave it right in front of your function so either way going that I hope that makes sense I'm gonna come back with one example this is kind of an optional thing for you I would definitely watch it we're gonna talk about the difference quotient one time all right well we got one more example we're gonna talk about the difference quotient and it's something that yes I just felt different like it's subtraction there's the difference quotient like the division you oh man new math guys are really unoriginal because you can call things just what they are and it's right like a difference of squares has a difference in has two squares here is a difference quotient because it's got a difference and we're dividing that's that's what a difference quotient it's is this thing now I'm going to tell you what it's used for after we after we solve it and then I'm going to tell you what the H does because it's it's gonna be quite interesting so know I catch anyone we do know here's what this does what this does in calculus once you're learning calc one right now kind of this is this would be precalculus but the idea that this is going to yield is a count one idea here's what we're doing the function itself is an output value do you see it like f of X is an output value do you understand that and f of X plus h would be an output value of whatever this happens to be so I'm subtracting out the values and dividing by the change between X and X plus h that little bit eh what that does is if I divide Y values they change in Y by a change in X do you remember what that what that gives like from forgetting from beginning algebra if I take this distance and divide by this distance that's a rise that's a run the HIV that run here what this does is gives a relationship for the slope of a line really doesn't look like slope it is sloping output to minus output 1 over the change between our inputs and that's exactly what that would be this is a slope formula in calculus 1 what you do is you say what if the H gets really small like what if this distance gets closer together like really close together what's going to happen well this these points would get really close together and one of the theorems and calculus is that if you take a limit as this distance approaches zero then this formula approaches the slope of a curve at a line and instantaneous rate of change very fascinating but that's we're going through I'm going to stop there because I'm not really going to teach you calculus right now just watch that one I do but you're going to see this right off the bat and how to evaluate it is incredibly important I'm not joking I've taught calculus now for my 15th year of teaching and I see the most mistakes right here people do not know how to deal with that you are not going to be those people so how to deal with this f of X plus h man can you see why I gave that to you in the second to last example on the last less whiteboard examples that we had it's because you're going to use it so here's how to appropriately deal with this difference quotient number one you're going to figure out what f of X plus h is that's just a valuation you're gonna keep f of X it's right there totally told to you you're gonna put them in here with parentheses and simplify it as much as you can your main goal is to cancel out that age somehow so let's go through it not quickly but but with some understanding of what evaluation means so what I need you to do here is figure out f of X plus h first figure out f of X second and then jam them together in this difference quotient so here is f of X and from X plus h I'm leaving this blank on purpose because I need you to know that this is no different from any other evaluation we've done it says take your function give it a blank space so let's see I'm gonna replace the independent variable X with something I'm gonna replace this infinite variable X with something so I'm gonna give it a blank space to do that in your head right now this and this need to be the same this x squared plus one says something I can plug in two squared plus one this blank space says something I can plug in two squared plus one I just let that link to illustrate that you can do anything you want to plug in 7 here or five here what a needs you to plug in is X plus h so I'm replacing the variable X for that blank space with X plus h here's a function f I'm replacing X with X plus h you know most people do this this is going to happen I see it all time they'll say Oh f of X plus h is super easy it's just I'm going to take the function f and just add H to it I'm gonna take this and add H that is not what is going on that would be like saying I'm gonna take three plus one it's four right and say I'm gonna do three squared plus one plus one it doesn't make sense like take your three plug it in and add one to the end we're not adding the H to the end what this means to do is replace the variable with one other thing X plus h non-separable I hope that's making sense that what we're doing is we're replacing this X with X plus h so we gave it a blank space and we just put the exposition in blank space simplify this thing now not later do it right now so when we simplify X plus h squared see X plus h times X plus h we distribute we've already done it x squared plus X H plus X H and then H squared we wrap it up combine some like terms x squared here's one x times h here's another x times h and here's an x squared so I'm going to get two x times h plus h squared this is exactly what f of x plus h is and then plus one so be very careful we if we distributed this we have this part right here was the X plus h squared that we multiply the X plus h on special stage we distributed and then that plus one has to be there so this is this whole part and that plus was at the end we just distributed this to here and then added that one I hope that makes sense to you now the difference quotient really relies on two different entities and puts them together in a difference it says now look at your f of X plus h so I need your f of X plus h I need you to simplify it first so distribute it combine some like terms this piece comes first we already know it and then we subtract f of X but we know those two things f of X plus h after expense of the Pfizer right here and f of X is this piece right here all the difference quotient does it says I want you to take those two entities I want you to put them together so we have to have the f of X plus h simplified there whirring subtract f of X I can't stress enough how important that parenthesis is because what's going to happen is because you're subtracting this you're subtracting every term in that f of X it's going to change all science I see it all the time man I've done some really long time the biggest thing is number one people don't know what to do f of X was H number two they forget the parentheses because they don't think it's important it's super important so let's let's put together the first term calls from f of X plus h and that's this big thing so x squared plus 2xh plus h squared plus one that is f of x plus h that what that's what we got that's what we simplified that's what that represents - the next piece is f of X of tank f of X plus h done - done f of X oh wait and the VIX has already gotten done it's given to me you doesn't work for that it's just an x squared plus falling can you see how easy it would be to accidentally do this did you see how easy it is to do that do you know that that's going to that's kind of drive you nuts because the main goal here the main goal is to cancel that H in order to do it every term up here must disappear must cancel out with the exception of those terms that have age so what's going to happen we figure out f of X plus h we've done that simplify it we've done it we put f of X plus h minus f of X in parenthesis and then we simplify all of our terms without H should cancel then we're going to factor H and cancel the H let's look at it so there's nothing to distribute this is x squared plus 2xh plus h squared plus 1 minus x squared and look minus 1 but this says that once you subtract everything in here once you subtract the x squared I want you subtract the 1 all over H our X Squared's are gone that's wonderful our ones are gone oh it's wonderful 1 minus 1 all that's left is 2x h plus h squared over H completely admit it everything on the numerator every term up there has an H in it we can factor them and what we know all about simplifying so if anything is multiplied together and has a common factor you can simplify it H divided by H is 1 1 times 2x plus H is just 2x plus h and you know what in the world does that mean and this is a problem with teaching math sometimes in the way we do it we don't tell you what that means we just say just do it because it's gonna be good for you here's what that means and this is going to be absolutely fascinating to you I hope it is I really hope it is some of you are going to be taking this class just because it's required of you but you have to realize that this class is entailed to get people ready for calculus here's what this does if you've done this and I've already told you this represents a slope between two points right and the H is the distance horizontally between those two points if I do something called a limit which is a way that we get closer and closer to some number in math we go oh what a little closer a little closer what happens if I let that H become really close to 0 if I let that edge right here become really close to zero then 2x plus something really close to 0 it's just gonna go closer and closer to X okay who cares well when you consider that this function right here is a parabola upward nobody shifted up one looks like that this is that parabola and as that H it's really little this becomes 2x so this gets closer and closer to 2x what that tells you is that depending on the x value if this is the slope and the slope simplified all the way down to here then you can find the slope at any point on this curve what I think so or curves head slopes they do but they're instantaneous and at every point change and so at any point that you want let's say let's say at two one two that would be four right so five five i plug in x equals two let's see x equals two maybe four plus one is five right there I'm we're not to scare or anything I'm just doing this for fun at this point if I plug into having yeah the number five that's the point two five if this is the slope at x equals two so at the point 2 comma 5 will be equal to that will have a slope of two x where your x value is 2 so 2 one more time the point on the curve is 2 comma 5 the slope at that point at the instant on that curve is going to be just plug in the x value it's based on X it is the slope I told you that's it so it is a soap plug into it's going to be 4 and you know what if you put that up there I saw it with 4 over 1 1 and up four that's the exact slope of that curve at that specific point before that it's gonna be steeper after that Masumi sorry before that's gonna be shallower after that's gonna be steeper but if that one point it's gonna be so lovely exactly for that is what you're doing this for so that's why we learned some of these compositions that's why we learn the difference quotient because the calculus one you do this time time again I have to give you one caution it's that right now that was fairly mathematic but easy if it is not easy compared to some things that we can have if you try to do difference quotient with something like this it's gonna be really really really challenging so square roots make these really challenging and fractions make them really really challenging to fortunately we have a lot of stuff in calculus that you don't have to deal with that with but you might run into some examples when you're doing something known then that have that so they're a little challenging I just want to give you the before warning that this is possible this is what I would expect is for you to be able to understand what this means and be able to put f of X plus h which we have f of X which was given and then simplified so I hope that makes sense I hope I've Illustrated why you would use this and what what we're kind of driving towards so that's that's the idea hope makes sense I will see you for another video when we talk about domain and we talk about range in how to phone you