                   # 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.  