8.4 Composite and Inverse Functions
The Composition of Functions
The composition of the function f with g is denoted by
fg
and is
defined by the equation
fg
( )
x
( )
=f g x
( )
( )
The domain of the composite function
fg
is the set of all x such that
1. x is in the domain of g and
2. g(x) is in the domain of f.
Forming Composite Functions
Example 1: Given f(x) = 5x + 2 and g(x) = 3x − 4, find (
fg
)(x)
and (
gf
)(x).
Inverse Functions
Let f and g be two functions such that
f(g(x)) = x for every x in the domain of g, and
g(f(x)) = x for every x in the domain of f.
The function g is the inverse of the function f, and is denoted by
f−1
(read “f-inverse”). Thus f(
f−1
(x))= x and
f−1
(f(x)) = x. The domain of f
is equal to the range of
f−1
and vice versa.
Verifying Inverse Functions
Example 2: Verify that each function is the inverse of the other:
f(x) = 6x and g(x) =
x
6
