Compound Functions
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:



Suppose h(x)
= x^{2} + 2x and j(x)
= + 2. What is j(h(4))? 

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:



Suppose f(x)
= 3x + 1 and g(x)
= . What is g(f(x))? 

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.