Identifying composite functions | Derivative rules | AP Calculus AB | Khan Academy

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 KH 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 + 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. 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 it with a G of x. So this is going to be equal to 1 + instead of the input being x, 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 instead of writing G of x there, I could write cosine of x.

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. Then we're going to 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 put it 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 sin of x + 1. And I also want to 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, all right, let's see. I have this S of x right over here. So what if I called that an F of x? So let's say I call that—let me actually, let me use a different variable so we don't get confused here. Let me use—let me call this U of x the S of x right over there. So this would be cosine of U of x + 1.

And so if we then defined another function as V of x being equal to cosine of whatever its input is + 1, 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 + 1. Let me write that down. So if we wrote V of U of x, which is S of x, if we did V of U of x, that is going to be equal to cosine of, instead of an x + one, it's going to be a U of x + 1.

And U of x—let me write this here—U of x is equal to sin of x; that's how we've set this up. So we could either write cosine of U of x + 1 or cosine of sin of x + 1, which was exactly what we had before. So this function G of x, if we say U of x is equal to sin of x, and V of x is equal to sin of x + 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 sin of x. We could define, let's say, a W of x to be equal to x + 1. And so then let's think about it. W of x, W of U of x, I should say, W of U—do 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 + 1, or just S of x + 1. So that's S of x + 1.

And then if we define a third function—let's say, let's see, I'll call it—let's call it H. I'm running out of variables. 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 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 + 1, where the U of x is S of x. But then we input that into W, so we got S of x + 1, and then we input 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 functions. For example, if I have some function—let me just clear this out. If I had some function f of x is equal to sin of x * sin 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—let me just a different color—I could say V of x is equal to sin 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 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 a review. Well, this is going to be— I take the U of x, takes the cosine of whatever is input, and now the input is V of x, which would be S of x.

And then if you did V of U of x, well, that would 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, all sorts of different combinations of how you can put functions together to create new functions.