Composite functions
Let ƒ(x) = x³, g (x) = sin x an...

Question

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.

Find ƒ(g(h( x))).

Accepted Answer

Welcome back, everyone. In this problem, we want to simplify the composite function a of b of c of x given that a of x is negative x cubed. B of x is the cosine of x, and the c of x is the square root of x. For our answer choices, a is the square root of the cosine of x. B is x cubed multiplied by the square root of the cosine of x. C is negative cosine of the square root of x all cubed. And the d is the negative cosine of x cubed. Now we want to simplify our composite function. That means we're putting one function into another. And to simplify our composite function, we evaluate the innermost function first. So for a of b of c of x, it means we're going to try and solve for b of c of x and then put that into a of whatever that result would be. So let's see if we can figure it out. Now, B of C of X means that anywhere we see our input X in B, we're going to put C of X. So B of X is going to be the cosine of not X, but the square root of x. Now that we have b of c of x, that tells us then that a of b of c of x is equal to a of the cosine of the square root of x. And now that tells us in a, wherever we see x, we're going to put the cosine of the square root of x. So that means a of x is now going to become negative the negative cosine of the square root of x cubed. And we can rewrite that as negative cosine of the square root of x all cubed. That would be our final answer. A of b of c of x. And when we look back in our answer choices, that would have been answer choice c. Thanks a lot for watching everyone. I hope this video helped.

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Composite functions
Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.
Find ƒ(g(h( x))).