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  • - [Instructor] 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 if 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,

  • and particularly 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 one plus x.

  • And let's say that we have g of x

  • is equal to, let's 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 gonna be equal to,

  • this is gonna be equal to one plus.

  • Instead of the input being x,

  • the input is g of x, so the output

  • is one plus g of x.

  • And g of x, of course, is cosine of x.

  • So instead of writing g of x there,

  • I could write cosine of x.

  • And one way to visualize this is,

  • I'm 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 then 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 if we can go the other way around.

  • Let's see if we can look at some type

  • of a function definition and say,

  • hey, can we express that as a composition

  • of other functions.

  • So let's start with, let's say

  • that I have a g of x

  • is equal to cosine of

  • sine of x plus one.

  • And I also wanna state,

  • there's oftentimes more than one way

  • to compose, 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, alright, well let's see,

  • I have this sine of x right over here.

  • So what if I called that an f of x?

  • So let's say I called that,

  • well actually let me use a different variable

  • so we don't get confused here.

  • Let me call this u of x,

  • the sine of x right over there.

  • So this would be cosine of u of x plus one.

  • And so if we then divided,

  • if we then defined another function as v of x

  • being equal to cosine of whatever its input is

  • plus one, well then this looks like the composition

  • of v and u of x.

  • Instead of v of x, if we did v of u of x,

  • then this would be cosine of u of x plus one,

  • let me write that down.

  • So if 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, instead of an x plus one,

  • it's going to be a u of x plus one.

  • And u of x, let me write this here,

  • u of x is equal to sine of x.

  • That's how we set this up.

  • So we can either write cosine of u of x plus one,

  • or cosine of sine of x plus one,

  • which was exactly what we had before.

  • And so this function, g of x,

  • is 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 one,

  • 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 one.

  • And so then, let's think about it.

  • W of x,

  • w of u of x, I should say,

  • w of u, I'll do the same color,

  • w of u of x is going to be equal to.

  • Now my input is no longer x, it's u of x,

  • so it's going to be a u of x plus one,

  • or just sine of x plus one.

  • So that's sine of x plus one.

  • And then if we define a third function,

  • let's say, let's see, I'll call it,

  • let's call it h; we're running out of variables,

  • although there are a lot of letters left.

  • So if I say h of x is just equal to the cosine

  • of whatever I input, so it's equal to the cosine of x,

  • well then h of w of u of x

  • is gonna 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 gonna 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 one,

  • where the u of x is sine of x,

  • but then we input that into w,

  • so we got sine of x plus one,

  • and then we inputed 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 wanna 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 the 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, it's 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.

  • If we were 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,

  • well that'd 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 have,

  • 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,