Warm-upWarm-up Determine and the domain of each Section 4.2 One-to-One and Inverse Functions

What did you find for: What do you think this implies ? What did you find for: What do you think this implies ? Motivation: Example of inverse functions These are the functions we use to convert between Celsius & Farenheit (Celsius to Fahrenheit) These are the functions we use to convert between Celsius & Farenheit (Celsius to Fahrenheit) (Fahrenheit to Celsius)

Section 4.2 One-to-One and Inverse Functions

Inverse Function as ordered pairs

Algebraic Example of an Inverse Notation : is used to represent the inverse of

1. Properties of Inverse of a function Definition: The inverse function of, is called and satisfies the property: and Definition: The inverse function of, is called and satisfies the property: and IMPORTANT: is NOT IMPORTANT: is NOT Domain of = Range of

2. Verify inverse functions State the Domain and Range of f and Example 1: Prove that f and are inverse functions

Example 2 : Verify that f and g are inverses of each other 2. More practice

3. Finding the Inverse of a Function Switch and solve 3. Finding the Inverse of a Function (Switch and solve) 1. Replace f(x) with y 2. Interchange x and y 3.Solve for y 4. Replace y with 1. Replace f(x) with y 2. Interchange x and y 3.Solve for y 4. Replace y with Exercises. Find the inverse of each function.

4. Domain of the Inverse of a Function State the domain and range for the function and its inverse. Domain of = Range of

5. Properties of the graph of Inverse Given the function: Find the inverse function and complete the table below. Given the function: Find the inverse function and complete the table below. x-intercepts y-intercepts vertical asymptotes horizontal asymptotes Sketch the graph of both on the same set of axes

6. Symmetry in the graphs Symmetry Symmetry: The graph of f -1 and f are symmetric with respect to the line y = x Points on the graph: If f contains the point (a,b) then f -1 contains the point (b,a) Points on the graph: If f contains the point (a,b) then f -1 contains the point (b,a)

6. Practice – Given a graph, sketch its inverse 1.Sketch inverse using symmetry about y = x. 2.Domain of f : 3.Range of f 4.Domain of f -1 5.Range of f –1 1.Sketch inverse using symmetry about y = x. 2.Domain of f : 3.Range of f 4.Domain of f -1 5.Range of f –1

7. Determine if a function is One-to-One Definition Definition: A function is one-to-one if for each y value there is exactly one x value (i.e. y values don’t repeat) Not one-to-oneIs one-to-one

7. Horizontal Line Test for Inverse Function Horizontal Line Test: f is a one-to-one function if there is no horizontal line that intersects the graph more than onceDefinition: f has an inverse that is a function if f passes the horizontal line test Definition : Domain-Restricted Function: A function’s domain can be restricted to make f one-to-one. Horizontal Line Test: f is a one-to-one function if there is no horizontal line that intersects the graph more than onceDefinition: f has an inverse that is a function if f passes the horizontal line test Definition : Domain-Restricted Function: A function’s domain can be restricted to make f one-to-one.

8. Finding the Inverse of a Domain-restricted Function Example: Restrict the domain of to make it one-to-one. Example: Restrict the domain of to make it one-to-one. Example: Restrict the domain of to make it one-to-one. Example: Restrict the domain of to make it one-to-one.