1. TOPIC 20.2 Composite and Inverse Functions

2. OBJECTIVES Form and Evaluate Composite Functions
Determine the Domain of Composite Functions
Determine If a Function Is One-to-One Using the Horizontal Line Test
Verify Inverse Functions
Sketch the Graphs of Inverse Functions
Find the Inverse of a One-to-One Function

3. Definitions Composite Function: One-to-One Function:
Given functions , the composite function, (also called the composition of ) is defined by provided is in the domain of
One-to-One Function:
A function is one-to-one, if for any values in the domain of
This definition suggests that a function is one-to-one if, for any two different input values (domain values), the corresponding output values (range values) must be different.
A function is one-to-one if, for any two range values
This definition says that if two range values are the same, then their corresponding domain values must be the same

4. Horizontal Line Test: If every horizontal line intersects the graph of a function at most once, then is one-to-one.
Inverse Function:
exists if and only if the function is one-to-one.

5. Composite Functions
Suppose are functions. For a number to be in the domain of must be in the domain of must be in the domain of Follow these two steps to find the domain of
Step 1. Find the domain of g.
Step 2. Exclude from the domain of g all values of x for which is not in the domain of f.
Example:
Find and its domain:
The domain of is The domain of cannot contain any values of that are not in this interval. In other words, the domain of is a subset of

6. Inverse Functions To find the inverse of a one-to-one function
replace
interchange the variables
solve for This is the function
Cancellation equations can be used to show whether two functions are inverses of each other.
Composition Cancellation Equations:

7. Graphs of Inverse Functions
Given the graph of a one-to-one function we can obtain the graph of by simply interchanging the coordinates of each ordered pair that lies on the graph of
Therefore, the graph of is a reflection of the graph of about the line
Example: Sketch the graph of and its inverse
Notice that if inverse functions
have any points in common, they must lie along the line