TOPIC 20.2 Composite and Inverse Functions

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

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

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.

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

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:

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