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) = 1/(2x-3)

Accepted Answer

welcome back. I'm so glad you're here. We are given this function H of X where H of X equals one divided by four. X minus nine. And we are told that H of X is equal to a composite function. It's the same as F of G of X. So those two are equal. What we need to figure out our what our what our our G of X and our F of X functions. If we know that H of X is a composite of F and G. So what we have to do is decompose our H of X. And how do we do that? What we want to take a look at H of X and see what operations are happening to our X here while X is being multiplied by four and then it's having nine subtracted from it. And then the last thing that will happen to it is that it will have one divided by it. And when we look at our composite function, we need to remember the order of which function goes inside the other. Well the G of X is going inside of the F of X. So the F of X operations need to happen last. What's the last operation happening to our X. It's having one divided by it. So for affects function we're going to look at that last operation and we're going to say it one divided by and replace that last operation with an X. Because in our composite function, that X will be replaced with our G of X and the G of X can be all the remaining operations happening to X in R. H of X. So that G of X will be the denominator four x minus nine. And we can check this by going ahead and doing our composite function again where F of G of X will be taking the G of X and sticking it inside the F of X. So we'd have one divided by from our F of X. And anywhere there's an X and F of X, we replace it with G of X. So the denominator will read for X minus nine and yes, that checks out that is equal to our H of X. We look at our answer choices and this matches with answer choice. A 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) = 1/(2x-3)