5 2 Inverse Functions and Their Representations Calculate

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5. 2 Inverse Functions and Their Representations ♦ Calculate inverse operations ♦ Identify one-to-one

5. 2 Inverse Functions and Their Representations ♦ Calculate inverse operations ♦ Identify one-to-one functions ♦ Find inverse functions symbolically ♦ Use other representations to find inverse functions Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Inverse Operations Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5

Inverse Operations Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 2

Reminder of the definition of a function Copyright © 2006 Pearson Education, Inc. Publishing

Reminder of the definition of a function Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 3

Example • Given y 2 = x, is y = f(x)? That is, is

Example • Given y 2 = x, is y = f(x)? That is, is y a function of x? Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 4

Idea Behind a One-to-One Function Copyright © 2006 Pearson Education, Inc. Publishing as Pearson

Idea Behind a One-to-One Function Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 5

Example • Given y = f(x) = |x|, is f 1 -1? Copyright ©

Example • Given y = f(x) = |x|, is f 1 -1? Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 6

Formal Definition of One-to-One Function Copyright © 2006 Pearson Education, Inc. Publishing as Pearson

Formal Definition of One-to-One Function Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 7

Given a 1 -1 function f Copyright © 2006 Pearson Education, Inc. Publishing as

Given a 1 -1 function f Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 8

Example of An Inverse Function Copyright © 2006 Pearson Education, Inc. Publishing as Pearson

Example of An Inverse Function Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 9

Example of An Inverse Function (con’t) Copyright © 2006 Pearson Education, Inc. Publishing as

Example of An Inverse Function (con’t) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 10

Example 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5

Example 2 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 11

Example 3 • Let f(x) compute the distance traveled in miles after x hours

Example 3 • Let f(x) compute the distance traveled in miles after x hours by a car with a velocity of 60 miles per hour. Explain what f -1 computes. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 12

Example 4 • Describe verbally the inverse of the statement. Then express both the

Example 4 • Describe verbally the inverse of the statement. Then express both the statement and its inverse symbolically. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 13

Let’s check Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5

Let’s check Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 14

Formal Definition of an Inverse Function Copyright © 2006 Pearson Education, Inc. Publishing as

Formal Definition of an Inverse Function Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 15

Using composition of functions verify that if 3 = f (x) x + 1

Using composition of functions verify that if 3 = f (x) x + 1 then Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 16

Using composition of functions verify that if then Copyright © 2006 Pearson Education, Inc.

Using composition of functions verify that if then Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 17

Procedure for finding Symbolic Representation of f -1 Copyright © 2006 Pearson Education, Inc.

Procedure for finding Symbolic Representation of f -1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 18

Why does f need to be 1 -1 to have an inverse function? Copyright

Why does f need to be 1 -1 to have an inverse function? Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 19

Evaluating an inverse function numerically x 1 2 3 4 5 f(x) – 5

Evaluating an inverse function numerically x 1 2 3 4 5 f(x) – 5 – 3 0 3 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 20

Evaluating an inverse function graphically f(2)=4 Copyright © 2006 Pearson Education, Inc. Publishing as

Evaluating an inverse function graphically f(2)=4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 21

Graphs of Functions and Their Inverses • The graph of f -1 is a

Graphs of Functions and Their Inverses • The graph of f -1 is a reflection of the graph of f across the line y = x Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 5 - 22

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