Home
/ Review
Topics
Composite
Functions
Given two functions f and
g,we can form a new function
as follows: start with
an input value x in the domain of g and find the corresponding output
value g(x). If the number g(x) is in the domain of f, we can then
find the output value of f at g(x). That is, we can find the value f(g(x)).
This process is called composition of functions. See figure 1 below.
Figure 1
In general,
for any two functions f and g, the composite function 
is
defined by
The
domain of 
is the set of all
numbers, x, in the domain of g for which g(x) is in the domain of
f.
Example
1
Suppose that
and
.
Find
a) 
b) 
c) 
Solution:
a)
b)
c)
Example
2
Given
the functions
find the domain of the
composite function 
.
Solution
To find
the domain of 
, we first notice that
the domain of g is
and the domain of f is
.
Thus, g(x) cannot equal
0. So we solve the equation
We must exclude -3 from
the domain of 
. The domain of
is
To check our answer, we can find the composite function
We see that 
is
undefined.
