Composition of Functions: Domains & De-Compositions

When composing functions, sometimes the output of one function can create issues as inputs to the other function.

As a result, you have to be careful with the domains and ranges of composite functions. And domains and ranges of compositions are an area which can generate trick questions. (Fair warning: You should expect at least one of these trick questions on the next exam.)

What is an example of finding the domain of a function composition?

• Given and g(x) = x − 2, find the domain of (f ∘ g)(x).

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• Given and g(x) = x − 2, find the domain of (g ∘ f )(x).

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Note: If your initial functions are just plain old polynomials, then their domains are "all x", and so will be the domain of the functions' composition. It's pretty much only if you're dealing with rational functions with denominators (where you can't divide by zero) or even-index radical functions like square roots (where you can't have a negative inside the radical) that the domain ever becomes an issue.

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Thus far, we've taken two functions and composed them together. Sometimes, they'll want you to go the other way.

What does it mean to decompose a function?

To decompose a function is to try to find two (or more) functions which, when composed, give you the function they'd given you to started with. There can be more than one valid answer to a given decomposition exercise. It is usually pretty obvious what functions they had in mind when they wrote the exercises. But any answer you come up with — any answer that works, of course — should be an acceptable solution.

• Given h(x) = (x + 1)² + 2(x + 1) − 3, determine two functions f (x) and g(x) which, when composed, generate h(x).

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Can there be more than one set of functions that compose to the same thing?

While it is usually pretty clear which two functions were composed to get a "Decompose this function" result, there is nothing that says that there must necessarily be only one set of functions that were composed. Any pair of functions which give the correct result will be valid mathematical answers, and should be acceptable to your grader as a correct solution — as long as you're not insultingly trivial about it.

In the previous exercise, I mentioned the fact that the author of that exercise had not multiplied things out, which made my life pretty easy. But I can find other solutions, especially if I do that multiplying and simplifying. What they had given me simplifies as:

Is there any way to decompose this? Sure! Here's an optional set of functions:

Now I'll compose them, to prove that they give h(x):

This is the same thing they'd given me in the first place; it looks different because I just composed to get the simplified form, where they'd given me the very-much-not simplified form.

As long as you show all your steps and reasoning (in this case, by multiplying out what they'd given me, simplifying, and working with that result, instead), you should get full credit.

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• Given , determine two functions f (x) and g(x) which, when composed, generate h(x).