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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 byThe 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 .













Example 2

 Given the functions

find the domain of the composite function .


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.