In Exercises 75-82, express the given function h as a composition...

Question

In Exercises 75-82, express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x). h(x) = ∛(x² – 9)

Accepted Answer

Welcome back. I'm so glad you're here. We are given the function H of X where H of X equals the cubed rut of X squared plus six. And we're told that H of X equals a composite function F of G of X. And we are asked to find F of X and G of X. So when we have a composite function F of G of X, we need to think about what goes inside of what and this G of X. The second one goes inside of the F of X. The first one listed G of X goes inside of F of X. Okay. Keeping that in mind for a moment, we're going to look back at our H of X function and we're going to see what is happening to our X. It's got several things happening. The first thing happening is that our X is being squared and then it's having six added to it and then it's having the cubed root of it taken. And so if our G of X is going inside of our F of X, the F of X is going to be the last operation that's happening to our function here. So the last operation happening to our X is having the cubed root of it taken. So for F of X that's going to be the cubed root of X. And then G of X. Because that goes inside of our F of X that's everything that's left that needs to happen to our X in R H of X function. So that's going to be X squared plus six. And we can check this. So to write our composite function, we start off writing our F. Of X. Function and we'll keep writing until we hit an X. So we can write the cubed root and there we've hit an X. Once we hit an X, we substitute in our G. Of X. So instead of the cubed root of X it's now the cubed root of X squared plus six. And yes, this matches our original H. Of X. Function. So we have correctly identified our F. Of X. And our G. Of X. We look at our answer choices and this matches with answer choice. D. Well done. We'll catch you on the next one.

In Exercises 75-82, express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x). h(x) = ∛(x² – 9)

Key Concepts

Function Composition

Cube Root Function

Quadratic Functions

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