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Master Function Operations & Build Composite Functions Easily
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Mar 20, 2025
We've already learned how to perform operations with polynomials. Now we'll learn how to find the sum, difference, product, and quotient of functions. Then we'll learn how to build composite functions. Chapters:
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0:00
so we're going to do chapter 6.1 which is operations with functions so we're going to talk about how to add subtract
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multiply and then divide functions pretty simple stuff but then when we get
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to composite functions that is something new that you going to learn today how to do composite function right so that is
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something new and it requires like you guys really pay attention now the first thing you want
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to learn is Basics notations right if you see me writing this on your any
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assignment right F + G of X that means I want you to add both functions that
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means I want you to do f ofx plus G of X okay this is simple step it's just
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notation that we learning difference is subtracting if you see this Fus G of X
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I'm actually asking you to do F ofx minus G of X and if you see this f * G
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that that means I'm asking you to multiply f of x and G of X and the last one is if you see F of G of X that means
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I want you to divide them but then you got to make sure that g of X is not equal to zero because a number or any
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function of a zero is on the fine we can't have that right it's Infinity we going to learn that in calculus okay now
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some examples that I want us to work on to get the hang of here is we have this here right I give you two functions f of
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x is X2 - 4 and then G of x is 2x + 1 and I want you to find F + G of X right
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wait wait so F + G of X so if I'm asking you to find this
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all I'm asking is add these two functions that's all we doing right F +
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G of X is combining this function by adding them right so that is basically
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x² - 4 + 2x + 1 wait wait wait
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still copy that over there all right copy a little faster right so here we just combining
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like terms so that gives me x² + 2x - 3 right so this is f plus G of
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X WR you're just adding them up all I'm doing is just adding things up I am not
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doing anything that you've never seen before so do not act surprised okay now
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the next one this is this minus
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what hey hey quiet down so down oh never mind so
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X2 - 4 + 2x + 1 I'm just adding them right so I'm just combining like turn
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and I have Fus G of X now I'm asking you guys to subtract
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that right so this is x² - 4 - and now you can put
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this in parenthesis 2x + 1 right so you're just again combining
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like terms we've done this before in the pols we learn how to add them subtract them multiply them so this is just
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composite functions they are called composite because you are putting them together you're creating functions out
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of two functions that are given to you so this is x² - 4 - 2x -1 and I'm just
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going to combine like terms be X2 - 2x - 5 all right that's f - G of
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X and then the next one is multiplying these two
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functions is f * G of X like this so this a
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multiplication so pretty much just change the signs kind of well here you change the sign all you're doing is just
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multiplying them when you see you just have to recognize the operation that has been asked of you right this operation
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here is adding them together this is subtracting them this is multiplying them next one will be dividing them
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right and you put in parentheses you know it's not a negative - 2x and it's minus 2x uh yes because if I don't put
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in parentheses you can a lot of people forget to like make sure that the negative changes the signed here and
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here some people forget that they just put Min - 2x + 1 and then you get the wrong answer so I put in parentheses to
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avoid any problem with my sign so I want to make sure I'm Distributing the negative evenly between the 2X and the
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one because it's going to change this into a negative and it's going to change this into a negative then we have F of g
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f f. G that means multiplying it right so that's x - 4 parentheses * 2x + 1 now
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this requires a little more work because we're going to foil right so it's X2 * 2x 2B that be 2x Cub right and X2 * 1
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that would be X2 then we have -4 * 2x - 8 x and then -4 * 1 - 4 now there
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is nothing else that I can do here because I have X cubed it cannot be
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combined with any other expression right then I have x² that stands alone because
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there's no other term with x² then I have 8x - 4 nothing else to do here so I
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am technically done with f * G of X then the very last one is subtract uh
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dividing so I have F over G so F over G of X that means I'm
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dividing both functions so I'm going to have x^2 -
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4 over 2x + 1 right so now here's
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there's a restriction that can be applied here because I said this here
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cannot be equal to zero right because any number of a zero is what undefined you can't have that it's undefined right
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so we want to make sure we want to make sure you want to make sure
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that 2x + 1 is not equal to zero we want to make sure that is true right which
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means 2x is not = to1 and X is not equal to1 / 2 because
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if x is = 1/2 it's going to turn this into a zero and we can't have that so as
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long as X is notal to1 / 2 this is f g of X
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this is how you divide F / G of X it's going to give you this as long as X is
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not equal to2 I'm adding this restriction for a purpose because we can't have zero in
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the denominator we can't have that so to make sure we avoid that we have to we have to tell whoever's looking at is
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that X not equal to2 that Mak say it's a restriction that we must add don't
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making sense okay so you see I have X2 -
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4 over 2x + 1 right you see how half this right so now this is f of x over G
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of X isn't it right that's true but the problem is this 2x + 1 x could be
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anything right X could be anything but the problem is since X can be
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anything if I replace this x by2 this is going to give me zero and
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that going to make this undefine so I have to add a restriction I have to say that 2x + 1 is not equal to zero right
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so X on the bottom can't be X canot
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be okay it cannot be 1/2 I have to specify that and if it is if it is then
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this is not true so it's undefined it's undefined so my thing is I'm just telling the general public that's
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watching hey F ofx over G of X is equal
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to this as long as X is not equal to this is called a domain you're going to
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see that probably in the next chapter we're going to cover a domain the domain of the function is the function the
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values for which the function is defined right so the function is defined this function will be defined as long as X is
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not equal to what 12 so I have to add that restriction
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because if I don't add it somebody consider that X is here and
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therefore that will be on the right for all the other functions there's nothing that will make this undef x + 2x - you
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can put in any value there it will it will stand right so then how do you solve it now you don't solve it that's
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it this is it that's it if x is2 the function is on theine we just
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have to make sure we let people know for for example let me give you another function if I give you a function like
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this right let's say you have a function 2/x right this is your function so you
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just have to add a restriction this is true as long as what as long as X doesn't equal z x not
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equal to zero that's all you're saying so this is all this is your final answer but you just have to make sure you tell
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somebody X can only equal to zero because you're going to have two over zero and that's undefine right or
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another example if I have -1 over x -1 why would be the Restriction
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here if I have this x cannot be equal to one X cannot be equal to one right
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because X - 1 cannot be equal to zero which mean x canot equal 1 so this is the Restriction that you have to add
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just to make sure right you're getting a phone call mind I'm still lost right
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now I went away okay so tell me where you lost because I want to address that
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specific thing that you L about how is it that you solve this
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because we keep saying that you have to add the Restriction okay I I'll I'll get to that the whole adding restriction
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thing got all right so let me show you
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something here right I don't know why so F of G of X
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you get this right that this is equal to x² - 4 over 2x + 1 right you agree to this
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right but now here's the thing you could stop here but the problem is this right with
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a rational function you know what a rational function is right what is a rational function rational function is
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something that makes sense essentially what what is it sense IR rational
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function is a function that has a denominator IR rational right rational
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okay so we have uh so this is a rational function here you have a denominator
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right now the problem is this in mathematics you can't have anything over
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zero that's not that's undefined right but the problem is with
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this function that I have here there could be a possibility that this will be equal to zero that's what I'm saying
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right so suppose I replaced X by-2 right if I replace X by -2 I'm going to get x2
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- 4 over 0 because 2 * -2 be1 and 1 + 1
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be zero so the reason why I'm talking about a restriction is this you have to
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once you write your function right you have to make sure that the denominator
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is not equal to zero so would you always replace X by 12 not necessarily so how do you know what to replace it that's
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where I'm going to get that makes the bottom one equal to zero what you want to do is this you want to say that 2x +
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1 you do not want 2x + 1 to be equal to Zer right that's what you want at all
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time in this specific case you do not want this to be equal to zero right why
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don't we want it to be equal to zero exactly so fine so that means now
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I'm going to solve for x I know that 2x + 1 can not be equal to Z I want to find the x value that's going to make this
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not equal to zero right so I'm going to solve it 2X notal
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1 and then X is not equal to -2 so I'm going to say f ofx over G of X is X2 - 4
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over 2x + 1 If X is not equal to1 that's the Restriction that's all we do okay WR
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we do not you just write write that down and say you do not want X to equal to2 because if x = 12 it makes the
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denominator zero which gives this uh the undefined t does that make sense now we
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good all right so now we're going to move on to composite functions right you
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want to write this down go ahead right now we're going to talk about composite functions now here I'm
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going to I'm going to need you guys really really pay attention because you've never seen this before okay now
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basically what is a composite function the word composite what does that mean in the dictionary together putting
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things together right composite is pretty much combining things right so with composite functions the result of
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one function are used to evaluate the other function right that's what happen
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when you have composite function you use the result of one function to evaluate another function so how is that going to
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happen first we need to know notations right so when I write this F Circle G of
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X I'm sure you've never seen this term before right what does that
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mean say it again f x so what does that
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mean so what does that mean so we're going to find that out what does that
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mean so it's called we can use the term there's two things that we
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can excuse me I have I still use that same notation
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confusing I don't understand your question can you write that down so what you saying so one of my functions
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is oh you know I'm not going to give you I get what you're saying that was a good
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one but I'm not going to give you to avoid any problems right so F of G of X
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so there's two ways to read this you can read it as f um
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round G of x or F Circle
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G of X that's what that means in terms of like how would you read this so
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f f of G of this F Circle G of X means F round G of X or F Circle G of X what
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does that mean it's not just something that is just like random is it means
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something so what it means is this you are finding right you are finding
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F of G of X and I will explain right so f r g of X is the equivalent of finding
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F of G of X and I will explain that so allow me to explain that okay so let me
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say let's say I have a function f ofx = 3x + 5 right let's assume that I
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have a function given 3x + 5 and I should find F of one what
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would you do one in for X you pluging one in for X
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right pluging one in for X so you it's going to give you a 3 * 1 + 5 you agree
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with me right everybody agrees on that we all good on that right now what if I ask you to find F of
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um let's say F of T what would you
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do put for X give you 3 t + 5 right all right now let me ask you to find F of G
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of X what would you do um put in G for X put in G for X
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right you be 3 * what g x g of X Plus what five that's
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basically what this means so if I put f g of X I'm asking
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you to find replace G of f x into F wherever you see a x value that's all
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I'm saying right so excuse me just replace it that's it just replace it and
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then now if you know G of X you can replace G of X by its actual value we're going to do some examples here and that
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will make sense right now look at this look at this one here so f ofx is given what is f
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ofx 3x - 1 right and then G of X is also given what is g of x x + 5 x + 5 right
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so now I want to find F round G of
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X right so what am I going to do here using 3x and put in 5 actually it's you
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go the other way around so first let's use like the sear rotation right f g of
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X will be me saying I'm finding what F of what GX G
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GX going step by step so F Circle G of X meaning I'm finding F of G of X and now
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to find it I'm going to do what I'm going to take this guy and I'm going to plug in where right here right so I'm
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going to go step by step first I'm going to go step by step I'm going to go this is 3 * what G of x - one right and then
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do I know G of X what is g of x x + 5 x + 5 so now that becomes 3 * x + 5 - 1
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right and then uh from here we can calculate right so that becomes
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3x + 15 - 1 and that is 3x+ 14 so when I'm asking for f of G of
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x f r g of X all I'm saying is this take G of X and plug that into F does that
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make sense so now I can go back I can I switch it up now what if I ask you to
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find so using the same example right 3x - 1 so f ofx is 3x
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-1 and G of X is x + 5 now I want you to
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find G round f of x what does that mean then yeah g m
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parentheses f pareses x x right that's what that means
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so now how would you do it you plug in so it would be so would be parentheses 3x - 1 + 5 yeah good but
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before we do that let's go step by step I'm going to replace X by what uh G I mean f of x so it's going to be F ofx
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right + 5 because all I'm doing is I'm taking this guy and I'm replacing it here right and then now I know what f of
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x is so it's what 3x -1 + 5 all right so that's it so that
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becomes 3x+ 4 right so G round f of
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x is = to 3x + 4 all right so that's basically what we doing in this chapter
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is called composite functions taking one function plugging that function into another one and then finding what it is
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so now let me give you another example and see how we're going to do this one so let me assume
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that f ofx
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is and then G of X is uh let's say x + one right so first
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I want you to find F round G X go ahead and and figure it out see
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what you get so let's find F GX using that method let's see what that's
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going to give you so f x is x² - 2x and
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G of X is x + 1 and I want to find F round G of X do we have work on this
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today I'm contemplating that so I want you to do this for
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reason so let's go ahead and find
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out all right so I want to find F of gfx so tell me what you how you do it
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uh I did M well I just skipped I just put the numbers in yeah go step by step
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okay so be what F of what first F of G of X all right and then and then it's F
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of G of x s so GX GX S right- 2 mhm time
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GX GX right and now we can replace this right we know what GX is what is GX so x
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+ 1 2 be x + 1 2 - 2 * x + all right so
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this will be x + 1 2 - 2 * x + 1
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right yes I so all you're doing again is just
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taking this gentleman and you going to plug it in whatever you see solving it you can go ahead and solve it right once
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you get here you can go ahead and solve it right finish it expand it X all right so go ahead and exp it is that marker
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like see if you can erase that I don't know if that's like is it yeah maybe all right well get the hand saner oh no use
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this wipes it's mostly you can use a marker let me see marker you can use another marker it's mostly
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temporary go ahe this is per so now what I want you to do oh pass
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that down it's also want to find right
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I have a question yes how do you determine whether it's going to be f g x
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i I gave it to you I I gave it to you so you you got to look up the way it's written right here if I have G
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round F ofx what am I asking for
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yet FX differ excuse me you the same problem in G of X vers F so I can see
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the difference I don't think I question she's asking you write both you write the same
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problem doing right now G yeah so now is what you now if I
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have g g r f ofx right look he just did f of x I just did F ofx right so now I'm
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g g I'm doing G Circle F so basically I'm asking for what G of
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what F ofx right so that's that's what I'm asking for which means now you going
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to take this guy and you going to plug it in with here so this will be G of f
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of x now will be what F ofx + one right G of f of x will be f of
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x + one because what I'm doing is I'm taking this guy and I'm going to plug it in here right that's what I'm doing in
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this case I'm taking that what it is and I'm replacing it by X in the other
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function right that's all I'm doing not not not like it's not that complicated and then
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now I know what FX is I'm going to replace it that becomes X2 - 2x + 1 all
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right so this is how you will do G of f ofx so now when we do this we do have to
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look at the Domain right so let me ask you a question here if you see this
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function um f ofx = x² - 2x what do you think the
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domain of the function is you know what the domain is right yeah do you have to make it equal zero now here it's a polom
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right is it two so all pols for every polinomial the domain is all real
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numbers is there anything that's going to make this on the fine no no not a single value that you plug in here is
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going to make this undefined right for all real numbers this is always going to be defined if you look at this function
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as as well g ofx = x + 1 right there is no value for which this does not exist
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so the domain of this function is all real numbers so if it was if it was x^2
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- 2x over x + 1 with the domain
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be1 over yeah it be all real numbers except negative 1 right but here the
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domain is all real numbers because there is no value for which this does not exist all right so now I also want to
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give you another example here so actually I want to sit on this for a little
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bitsu what my desk is excuse me
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huh yeah I'm going to yeah I'm going to give you some examples so let's say we have F
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ofx right excuse me Mr salami yes whenever it's convenient I have U
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something for Patrick um here in the front office all right thank I that was your birthday
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present so the question is saying
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right yeah so let's say we have this and I'll
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say
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fine f f Circle G of
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X also find G Circle f of x and then I
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also want to find F Circle f of x right so let's let's do this three problem and
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then the next question is if they exist
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right he got Aon oh you s so find these three if they exist so the first thing
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you want to do is we're going to look at each one of them right if I look here I already know that the domain of this
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function will be all real numbers there is no values for which this is undefined is there is there any value for which
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this is not going to be defined this no exist not right because for any value that you put in here you're going to be
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able to find it right here anything that you put in here you're going to be able to find it right x - 6 if you put in any
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number from negative Infinity to positive Infinity is not going to make this nonexistent
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nonexistent I mean right same thing with this one right so that's that now if I
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want you to find why you all playing so much down there this is serious stuff I don't want to hear tomorrow I don't know
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how to do this nobody's taking notes up so I don't want to hear I'm serious yeah
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yeah if I don't want to hear if you can do the homework doing on you not me so
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now I want to find F G of X right so if I want to find F G of
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X basically again this right is f of G of
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X that's what I'm finding F of gfx so what I'm doing now
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is I'm taking this guy and I'm replacing it here that's all I'm doing and then
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I'm finding the domain of that function so that's going to be G ofx
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s + 2 which is x - 6
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2+ two right and then I I won't mind if you stop here you don't need to expand
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it that's fine you can stop here right sense I'm tell you this today this makes
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sense right now but tomorrow I so now G
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FX right is basically G of f ofx X right so what you're doing
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is you're taking F ofx and now you plugging in into G that's what you're
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doing now right I'm taking this guy this guy this whole thing and it's going to be plugged into whatever I see an x
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value okay so that's going to be pretty much f
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ofx - 6 and now I know what f ofx is I'm going to replace it so that is X2 + 2 -
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6 now I don't stop here for this one because I I know I can combine two and six that gives me x^2 - 4 right and now
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lastly I have X yeah now I'm I'm doing F round F
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ofx right so I'm doing F round fation I'm plugging the same equation so
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I'm taking this guy right and I'm plugging into himself right all right you're pluging that function into itself
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so you basically doing F FX right so it's going to be
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FX squ + 2 because I'm taking F ofx and I'm plugging it back to
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itself so that's going to be X2 + 2 the whole thing squ + 2 right
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so you taking F and you inserting x f ofx back into itself that's that's all
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we're doing here right you're taking one and you reinserting it into itself so that gives you f of x² + 2 which is X2 +
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2 2 + 2 so okay x 4 you can stop here
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once you get that once you get this point you don't have to do anything else just stop right I don't need you to expand it because you'll be useless I
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just need to know that you know how to actually get to that right I don't really need the rest it's just fluff
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okay so what I want to I'm going to give you some work to do and then um tomorrow
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we're going to continue on with like this composite functions but we actually going to have like brackets and we're
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going to figure out how to like use that cuz I don't want to do that today so I'm just going to give you some
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Basics this