Copyright © 2011 Pearson Education, Inc. Logarithmic Functions Chapter 11

Copyright © 2011 Pearson Education, Inc. Inverse Functions Section 11.1

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 3 Copyright © 2011 Pearson Education, Inc. Definition of an Inverse of a Function Fahrenheit/Celsius Relationship Observations

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 4 Copyright © 2011 Pearson Education, Inc. Definition of an Inverse of a Function Fahrenheit/Celsius Relationship Observations Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 5 Copyright © 2011 Pearson Education, Inc. Definition of an Inverse of a Function Fahrenheit/Celsius Relationship Observations Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 6 Copyright © 2011 Pearson Education, Inc. Definition of an Inverse of a Function Fahrenheit/Celsius Relationship There are two key observations we can make about g −1 1. g −1 sends outputs of g to inputs of g. For example, g sends the input 0 to the output 32 and g −1 sends 32 to 0 (see Figs. 1 and 2). Using symbols, we write We say that these two statements are equivalent, which means that one statement implies the other and vice versa. Observations Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 7 Copyright © 2011 Pearson Education, Inc. Definition of an Inverse of a Function Fahrenheit/Celsius Relationship 2.g −1 undoes g. For example, g sends 0 to 32 and g −1 undoes this action by sending 32 back to 0. For an invertible function f, the following statements are equivalent: f (a) = b and f −1 (b) = a In words: If f sends a to b, then f −1 sends b to a. If f −1 sends b to a, then f sends a to b. Property Observations Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 8 Copyright © 2011 Pearson Education, Inc. Definition of an Inverse of a Function Evaluating an Inverse Function Let f be an invertible function where f (2) = 5. Find f −1 (5). Since f sends 2 to 5, we know that f −1 sends 5 back to 2. So, f −1 (5) = 2. Some values of an invertible function f are shown in the table on the next slide. 1. Find f (3). 2. Find f −1 (9). Example Solution Example

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 9 Copyright © 2011 Pearson Education, Inc. Definition of an Inverse of a Function Evaluating f and f −1 1. f (3) = Since f sends 2 to 9, we conclude that f −1 sends 9 back to 2. Therefore, f −1 (9)=2. The −1 in “f −1 (x)” is not an exponent. It is part of the function notation “ f −1 ”—which stands for the inverse of the function f. Here, we simplify 3 −1 and use the values of f shown in the table to find f −1 (3): Solution Warning

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 10 Copyright © 2011 Pearson Education, Inc. Definition of an Inverse of a Function Evaluating f and f −1 The graph of an invertible function f is shown. 1.Find f (2). 2.Find f −1 (5). Warning Continued Example

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 11 Copyright © 2011 Pearson Education, Inc. Definition of an Inverse of a Function Evaluating f and f −1 1. The blue arrows in the figure show that f sends 2 to 3. So, f (2) = The function f sends 4 to 5. So, f −1 sends 5 back to 4 (see the red arrows). Therefore, f −1 (5) = 4. Solution

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 12 Copyright © 2011 Pearson Education, Inc. Definition of an Inverse of a Function Finding Input-Output Values of an Inverse Function Let 1.Find five input–output values of f −1. 2.Find f −1 (8). We begin by finding input–output values of f (see the table). Since f −1 sends outputs of f to inputs of f, we conclude that f −1 sends 16 to 0, 8 to 1, 4 to 2, 2 to 3, and 1 to 4. We list these results from the smallest to the largest input in the table. Example Solution

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 13 Copyright © 2011 Pearson Education, Inc. Definition of an Inverse of a Function Finding Input-Output Values of an Inverse Function 2.From the table, we see that f −1 sends the input 8 to the output 1, so f −1 (8) = 1. If f is an invertible function, then f −1 is invertible, and f and f −1 are inverses of each other. Solution Continued Properties

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 14 Copyright © 2011 Pearson Education, Inc. Graphing Inverse Functions Comparing the graphs of a Function and Its Inverse Sketch the graphs of f (x) = 2 x, f −1, and y = x on the same set of axes. Solution Example

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 15 Copyright © 2011 Pearson Education, Inc. a Graphing Inverse Functions Comparing the graphs of a Function and Its Inverse ExampleSolution Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 16 Copyright © 2011 Pearson Education, Inc. For an invertible function f, the graph of f −1 is the reflection of the graph of f across the line y = x. For an invertible function f, we sketch the graph of f −1 by the following steps: 1. Sketch the graph of f. 2. Choose several points that lie on the graph of f. 3. For each point (a, b) chosen in step 2, plot the point (b, a). Graphing Inverse Functions Comparing the graphs of a Function and Its Inverse Property Process

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 17 Copyright © 2011 Pearson Education, Inc. 4.Sketch the curve that contains the points plotted in step 3. Let f (x) = 1/3x − 1. Sketch the graph of f, f −1, and y = x on the same set of axes. We apply the four steps to graph the inverse function: Step 1. Sketch the graph of f. Graphing Inverse Functions Graphing an Inverse Function Process Continued Example Solution

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 18 Copyright © 2011 Pearson Education, Inc. Step 2. Choose several points that lie on the graph of f : (−6,−3), (−3,−2), (0,−1), (3, 0), and (6, 1). Step 3. For each point (a, b) chosen in step 2, plot the point (b, a): We plot (−3,−6), (−2,−3), (−1, 0), (0, 3), and (1, 6) in the figure. Graphing Inverse Functions Graphing an Inverse Function Solution Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 19 Copyright © 2011 Pearson Education, Inc. Step 4. Sketch the curve that contains the points plotted in step 3: The points from step 3 lie on a line. Graphing Inverse Functions Graphing an Inverse Function Solution Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 20 Copyright © 2011 Pearson Education, Inc. Revenues from digital camera sales in the United States are shown in the table for various years. Let r = f (t) be the revenue (in millions of dollars) from digital camera sales in the year that is t years since A reasonable model is f (t) = 0.73t Find an equation of f −1. 2. Find f (10). What does it mean in this situation? Finding an Equation of the Inverse of a Model Example

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 21 Copyright © 2011 Pearson Education, Inc. 3. Find f −1 (10). What does it mean in this situation? 4. What is the slope of f ? What does it mean in this situation? 5.What is the slope of f −1 ? What does it mean in this situation? 1. Since f sends values of t to values of r, f −1 sends values of r to values of t Finding an Equation of the Inverse of a Model ExamplExample Continued Solution

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 22 Copyright © 2011 Pearson Education, Inc. To find an equation of f −1, we want to write t in terms of r. Here are three steps to follow to find an equation of f −1 : Step 1. We replace f (t) with r : r = 0.73t Step 2. We solve the equation for t: Finding an Equation of the Inverse of a Model ExamplSolution Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 23 Copyright © 2011 Pearson Education, Inc. Step 3. Since f −1 sends values of r to values of t, we have f −1 (r) = t. So, we can substitute f −1 (r ) for t in the equation t = 1.37r − 2.11: Check that the graph of f −1 is the reflection of f across the line y = x. Finding an Equation of the Inverse of a Model ExamplSolution Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 24 Copyright © 2011 Pearson Education, Inc. 2. f (10) = 0.73(10) = Since f sends values of t to values of r, this means that r = 8.84 when t = 10. According to the model f, the revenue will be about $8.8 million in Finding an Equation of the Inverse of a Model ExamplSolution Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 25 Copyright © 2011 Pearson Education, Inc. 3. f −1 (10) = 1.37(10) − 2.11 = Since f −1 sends values of r to values of t, this means that t = when r = 10. According to the model f −1, the revenue will be $10 million in The slope of f (t) = 0.73t is This means that the rate of change of r with respect to t is According to the model f, the revenue increases by $0.73 million each year. Finding an Equation of the Inverse of a Model ExamplSolution Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 26 Copyright © 2011 Pearson Education, Inc. 5.The slope of f −1 (r) = 1.37r −2.11 is This means that the rate of change of t with respect to r is According to the model f −1, 1.37 years pass each time the revenue increases by $1 million. Finding an Equation of the Inverse of a Model ExamplSolution Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 27 Copyright © 2011 Pearson Education, Inc. Find the inverse of an invertible model f, where p = f (t). 1. Replace f (t) with p. 2. Solve for t. 3. Replace t with f −1 (p). Finding an Equation of the Inverse of a Model Three-Step Process for Finding the Inverse Function Process

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 28 Copyright © 2011 Pearson Education, Inc. Find the inverse of f (x) = 2x – 3. Step 1. Substitute y for f (x): y = 2x − 3 Step 2. Solve for x: Finding an Equation of the Inverse of a Function That is Not a Model Finding the Inverse of a Function that Is Not a Model Example Solution

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 29 Copyright © 2011 Pearson Education, Inc. Step 3. Replace x with f −1 (y): Step 4. When a function is not a model, we usually want the input variable to be x. So, we rewrite the equation in terms of x: To verify our work, we use ZStandard followed by ZSquare to check that the graph of f −1 is the reflection of the graph of f across the line y = x. Finding an Equation of the Inverse of a Function That is Not a Model Finding the Inverse of a Function that Is Not a Model ExampleSolution Continued

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 30 Copyright © 2011 Pearson Education, Inc. Let f be an invertible function that is not a model. To find the inverse of f, where y = f (x), 1. Replace f (x) with y. 2. Solve for x. Finding an Equation of the Inverse of a Function That is Not a Model Finding the Inverse of a Function that Is Not a Model ExampleSolution Continued Process

Lehmann, Elementary and Intermediate Algebra, 1 ed Section 11.1Slide 31 Copyright © 2011 Pearson Education, Inc. 3. Replace x with f −1 (y). 4.Write the equation of f −1 in terms of x. If each output of a function originates from exactly one input, we say that the function is one-to-one. A one-to-one function is invertible. Finding an Equation of the Inverse of a Function That is Not a Model Finding the Inverse of a Function that Is Not a Model ExampleProcess Continued Definition