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# 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.