A compound function is a function that operates on another function. A compound function is written as nested functions, in the form f(g(x)). To evaluate a compound function, first evaluate the internal function, g(x). Next, evaluate the outer function at the result of g(x). Work with the inner parentheses first and then the outer ones, just like in any other algebraic expression. Try the following example:

To evaluate this compound function, first evaluate h(4):

Now plug 24 into the definition of j:

It is important that you pay attention to the order in which you evaluate the compound function. Always evaluate the inner function first. For example, if we had evaluated j(x) before h(x) in the above question, you would get a completely different answer:

Here’s a slightly more complicated example, in which you are not given a specific point to evaluate a compound function:

In a case where you are not given a constant at which to evaluate a compound function, you should simply substitute the definition of f(x) as the input to g(x). This situation is exactly the same as a regular equation being evaluated at a variable rather than a constant.