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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,