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
as 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 of the compound function to evaluate
:



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

When 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.