6.2 One-to-One Functions; Inverse Functions MAT 204 SPRING 2009 6.2 One-to-One Functions; Inverse Functions In this section, we will study the following topics: One-to-one functions Finding inverse functions algebraically Finding inverse functions graphically Verifying that two functions are inverses, algebraically and graphically

MAT 204 SPRING 2009 Inverse Functions Example*: Solution:

Inverse Functions Notice, for each of the compositions f(g(x)) and g(f(x)), the input was x and the output was x. That is because the functions f and g “undo” each other. In this example, the functions f and g are __________ ___________.

Definition of Inverse Functions MAT 204 SPRING 2009 Definition of Inverse Functions The notation used to show the inverse of function f is f -1 (read “f-inverse”). Definition of Inverse Function f -1 is the inverse of function f f(f -1(x)) = x for every x in the domain of f -1 AND f -1(f(x))=x for every x in the domain of f The domain of f must be equal to the range of f -1, and the range of f must be equal to the domain of f -1. We will use this very definition to verify that two functions are inverses.

Watch out for confusing notation. MAT 204 SPRING 2009 Watch out for confusing notation. Always consider how it is used within the context of the problem.

MAT 204 SPRING 2009 Example Use the definition of inverse functions to verify algebraically that are inverse functions.

A function f has an inverse if and only if f is ONE-TO-ONE. MAT 204 SPRING 2009 One-to-one Functions Not all functions have inverses. Therefore, the first step in any of these problems is to decide whether the function has an inverse. A function f has an inverse if and only if f is ONE-TO-ONE. A function is one-to-one if each y-value is assigned to only one x-value.

One-to-one Functions Using the graph, it is easy to tell if the function is one-to-one. A function is one-to-one if its graph passes the H____________________ L_________ T__________ ; i.e. each horizontal line intersects the graph at most ONCE. one-to-one NOT one-to-one

Example Use the graph to determine which of the functions are one-to-one.

Finding the Inverse Function Graphically To find the inverse GRAPHICALLY: Make a table of values for function f. Plot these points and sketch the graph of f Make a table of values for f -1 by switching the x and y-coordinates of the ordered pair solutions for f Plot these points and sketch the graph of f -1 The graphs of inverse functions f and f -1 are reflections of one another in the line y = x.

Finding the Inverse Function Graphically (continued) Example: y=x f(x)=3x-5 x f(x) -5 1 -2 2 3 4 x f-1(x) -5 -2 1 2 4 3

Which of the following is the graph of the function below and its inverse?

Finding the Inverse Function Algebraically To find the inverse of a function ALGEBRAICALLY: First, use the Horizontal Line Test to decide whether f has an inverse function. If f is not 1-1, do not bother with the next steps. Replace f(x) with y. Switch x and y Solve the equation for y. Replace y with f -1(x).

Finding the Inverse Function Algebraically Example Find the inverse of each of the following functions, if possible. 1.

Finding the Inverse Function Algebraically Example Find the inverse of each of the following functions, if possible. 2.

Finding the Inverse Function Algebraically Example (cont.) 3.

Finding the Inverse Function Algebraically Example (cont.) 4.

Java Applet Click here to link to an applet demonstrating inverse functions.

MAT 204 SPRING 2009 End of Section 6.2