Composite functions are formed by taking the output of one function and using it as the input of another function. This is shown notationally as f(g(x)), where the result of g(x) is used as the input for f. Changing the order of the functions changes the result, as the output of the inner function determines the input to the outer function. Examples show evaluating composite functions by substituting the output of the inner function into the outer function and simplifying.

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 Given the composite function 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. Try These! For the functions find

7. Solutions! For the functions find