1. Composite Functions

2. What Are They?
Composite functions are functions that are formed from two functions f(x) and g(x) in which the output or result of one of the functions is used as the input to the other function. Notationally we express composite functions as
In this case the result or output from g becomes the input to f.

3. Example 1 Given the composite function Replace g(x) with x+2
Replace the variable x in the f function with x+2
Expand

4. Example 2 the composite function Given
The result of the function h becomes the input to k
Replace the variable x in k(x) with
Simplify

5. Example 2 Con’t.
Now see what happens when we take the same two functions and reverse the order of the composition.
The composite function
Notice, the result here is not the same as the previous result. This is usually the case with composite functions. Changing the order of the composition (changing which function is the “inner” function and which is the “outer” function) usually changes the result.

6. Problem 1 For the functions find (click mouse to see answer)

7. Breaking Composite Functions Apart
There are instances when we want to take a composite function and break it into its component parts. In this case we’ll be looking for an “inner” function and an “outer” function. To help you find the inner function look for expressions in parentheses, or under radical signs or in denominators.

8. Example 3 Break the composite function into two smaller functions
so that
Inner part
Outer part

9. Problem 2
Break the given function h(x) into components f(x) and g(x) such that
h(x) =f (g(x))
(click mouse to see answer)
(click mouse to see answer)