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# Identifying composite functions

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.C (LO)
,
FUN‑3.C.1 (EK)

## Video transcript

we're going to do in this video is review the notion of composite functions and then build some skills recognizing how functions can actually be composed if you've never heard of the term composite functions or the first few minutes of this video look unfamiliar to you I encourage you to watch the algebra videos on composite functions on Khan Academy the goal of this one is to really be a little bit of a practice before we get into some skills that are necessary in calculus in particular the chain rule so let's just review what a composite function is so let's say that I have let's say that I have f of X f of X being equal to 1 plus X and let's say that we have G of X is equal to say G of X is equal to cosine of X so what would F of G of X be F of G of X and I encourage you to pause this video and try to work it out on your own well one way to think about it is the input into f of X is no longer X it is G of X so everywhere where we see an X in the definition of f of X we would replace with the G of X so this is going to be equal to this is going to be equal to 1 plus instead of the input being X if the input is G of X so the output is 1 plus G of X and G of X of course is cosine of X so sort of writing G of X there I could write cosine of X and one way to visualize this is I'm going putting my X into G of X first so X goes into the function G and it outputs G of X and then we're gonna take that output G of X and then input it into f of X or input it into the function f I should say we input into the function f and that is going to output f of whatever the input was and the input is G of X G of X so now with that review out of the way let's see we can go the other way around let's see if we can look at some type of a function definition say hey can we express that as a composition of other functions so let's start with let's say that I have G of X is equal to cosine of sine of X plus 1 and I also want to state there's oftentimes more than one way to compose a or to construct a function based on the composition of other ones but with that said pause this video and say hey can i express G of X as a composition of two other functions let's say an F and an H of X so there's a couple of ways that you could think about it you could say all right let's see I have the sine of X right over here so what if I call that an f of X so let's say I call that well let me actually only use a different variable so we don't get confused here let me use let me call this U of X the sine of X right over there so this would be cosine of U of X plus 1 and so if we then divided if we then define another function as V of X being equal to cosine of whatever its input is plus 1 well then this looks like the composition of V and U of X instead of V of X if we did D of U of X then this would be cosine of U of X plus 1 let me write that down so this we wrote V of U of X which is sine of X if we did V of U of X that is going to be equal to cosine of a set of an X plus 1 it's going to be a U of X plus 1 and U of X let me write this here U of X is equal to sine of X that's how we've set this up so we could either write cosine of U of X plus 1 or cosine of sine of X plus 1 which was exactly what we had before and so this function G of X if we say U of X is equal to sine of X if we say U of X is equal to sine of X and V of X is equal to cosine of X plus 1 then we could write G of X as the composition of these two functions now you could even make it a composition of three functions we could keep u of X to be equal to sine of X we could define let's say a w of X to be equal to X plus 1 and so then let's think about it W of X W of U of X I should say W of U the same color W of U of X is going to be equal to now my input is no longer X it's a U of X so it's going to be a U of X plus 1 or just sine of X plus 1 so that's sine of X plus 1 and then if we define a third function let's say as we call it let's call it h I'm running out of variables well there's a lot of letters left so if I say H of X is just equal to the cosine of whatever our input so it's equal to the cosine of X well then H of W of U of X is going to be G of X let me write that down H of W of U of X U of X is going to be equal to remember H of X takes the cosine of whatever its input is so it's going to take the cosine now its input is W of U of X we already figured out W of U of X is going to be this business so it's going to be sine of X plus 1 where the U of X is sine of X but then we input that into W so we got sine of X plus 1 and then we inputted that into H to get cosine of that which is our original expression which is equal to G of X so the whole point here is to appreciate how to recognize compositions of functions now I want to stress it's not always going to be a composition of a function for example if I have some function let me just clear this out if I had some function f of X is equal to cosine of X times sine of X it would be hard to express this as a composition of functions but I can represent it as a product of functions for example I could say cosine of X I could say U of X is equal to cosine of X and I could say V of X which is a different color I could say V of X is equal to sine of X and so here f of X wouldn't be the composition of U and V it would be the product f of X is equal to U of x times V of X G of X if we were to take the composition if we were to say U of V of X pause the video think about what that is and that's a little bit of review well this is going to be I take U of X takes the cosine of whatever is input and now the input is V of X which would be sine of X sine of X and then if you did V of U of X will that be the other way around it would be sine of cosine of X but anyway this is once again just to help us recognize hey do i when I look at an expression or a function definition am i looking at products of functions am I looking at compositions of functions sometimes you're looking at products of compositions or quotients of compositions all sorts of different combinations of how you can put functions together to create new functions
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