After learning about the definition of a function we learned about how to evaluate a function at a real number. Recalling how this is defined, if we want to evaluate a function f(x) at x = 5, we would replace all occurrences of x with 5, and simplify. For composite functions, instead of replacing the independent variable, usually x, with a number, we replace it with a function.
Definition and notation
Given two functions, f and g, the composite function, denoted , is a function where .
The domain of a composite function is the collection of x-values in the domain of g such that g(x) is in the domain of f.
Example: Find the domain of
We start by noting that the domain of g(x) is . Now we want to know for what values of x is g(x) = -1. So we solve: . Solving this equation we find that g(-4) = -1. So -4 must be removed from the domain of g to result in th domain of . To finish the problem: the domain of