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) = (3x − 1)^4

Accepted Answer

Welcome back. I'm so glad you're here. We are given the function H of X where H of X equals two. X plus three. All raised to the fifth power. We are told that H of X equals a composite function F of G of X. We are asked to find F of X and G of X. So let's look at our composite function for a moment and recall which goes inside of which function? The second one listed here, G of X goes inside of the first one, the F of X. So G of X goes inside of F of X. Hold on to that for a moment and we'll come back to it. Now. Let's take a look at our H of X function and see what's happening to the X. Well, the first thing that happens to the X is that it gets multiplied by two And then it has three added to it and then it will be raised to the 5th power. Okay, So if G of X goes inside of F of X, then F of X is what's going to happen to X last. So looking at our H of X function, what happens to the X value last? The last thing is that it gets raised to the fifth power. So F of X is X raised to the fifth power. Then G of X gets substituted in for X in R. F of X. And so G of X is going to be everything that's left that has to happen to that X. So G of X is going to be two X plus three. And we can check this, let's do our composite function. So for our composite function we start off writing our F of X. And we stop as soon as we hit an X. And well in this case we hit an X immediately. And whenever we have an X we're going to substitute G of X. Ok. So instead of X to the fifth it's now going to be two X plus three and now resume writing our F of X now two X plus three raised to the fifth power. And yes, this does match our H of X. 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 B. Well done. We'll catch you on the next one.

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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) = (3x − 1)^4

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