Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Section 11.2 Inverse Functions Copyright © 2011 Pearson Education, Inc. Definition of an Inverse of a Function Fahrenheit/Celsius Relationship Observations Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 2 Definition of an Inverse of a Function Fahrenheit/Celsius Relationship Observations Continued Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 3 Definition of an Inverse of a Function Fahrenheit/Celsius Relationship Observations Continued Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 4 Definition of an Inverse of a Function Fahrenheit/Celsius Relationship Observations Continued −1 There are two key observations we can make about g −1 1. g 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. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 5 Definition of an Inverse of a Function Fahrenheit/Celsius Relationship Observations Continued −1 2. g undoes g. For example, g sends 0 to 32 and g undoes this action by sending 32 back to 0. −1 Property For an invertible function f, the following statements −1 are equivalent: f (a) = b and f (b) = a −1 −1 In words: If f sends a to b, then f sends b to a. If f sends b to a, then f sends a to b. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 6 Definition of an Inverse of a Function Evaluating an Inverse Function Example −1 Let f be an invertible function where f (2) = 5. Find f (5). Solution Since f sends 2 to 5, we know that f −1 f (5) = 2. −1 sends 5 back to 2. So, Example Some values of an invertible function f are shown in the table on the next slide. −1 1. Find f (3). 2. Find f (9). Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 7 Definition of an Inverse of a Function Evaluating f and f −1 Solution 1. f (3) = 27. 2. Since f sends 2 to 9, we conclude −1 that f sends 9 back to 2. Therefore, −1 f (9)=2. Warning −1 The −1 in “f (x)” is not an exponent. It is part of the function notation “ f −1”—which stands for the inverse of the function f . Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 8 Definition of an Inverse of a Function Evaluating f and f −1 Example The graph of an invertible function f is shown. 1. Find f (2). −1 2. Find f (5). Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 9 Definition of an Inverse of a Function Evaluating f and f −1 Solution 1. The blue arrows in the figure show that f sends 2 to 3. So, f (2) = 3. 2. The function f sends 4 −1 to 5. So, f sends 5 back to 4 (see the red arrows). −1 Therefore, f (5) = 4. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 10 Definition of an Inverse of a Function F i n d i n g I n p u t - O u t p u t Va l u e s o f a n I n v e r s e F u n c t i o n Example x 1 Let f x 16 . 2 −1 1. Find five input–output values of f . −1 2. Find f (8). Solution x f ( x) 0 16 1 8 2 4 3 2 4 1 We begin by finding input–output values of f (see the −1 table). Since f sends outputs of f to inputs of f , −1 we conclude that f 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. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 11 Definition of an Inverse of a Function F i n d i n g I n p u t - O u t p u t Va l u e s o f a n I n v e r s e F u n c t i o n Solution Continued −1 2. From the table, we see that f sends the input 8 to the output 1, −1 so f (8) = 1. Properties If f is an invertible function, then −1 • f is invertible, and −1 • f and f are inverses of each other. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 12 Graphing Inverse Functions Comparing the graphs of a Function and Its Inverse Example Sketch the graphs of f (x) = same set of axes. 2x , −1 f , and y = x on the Solution Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 13 Graphing Inverse Functions Comparing the graphs of a Function and Its Inverse Solution Example Continued a Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 14 Graphing Inverse Functions Comparing the graphs of a Function and Its Inverse Property −1 For an invertible function f , the graph of f is the reflection of the graph of f across the line y = x. Process For an invertible function f, we sketch the graph of −1 f 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). Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 15 Graphing Inverse Functions Graphing an Inverse Function Process Continued 4. Sketch the curve that contains the points plotted in step 3. Example −1 Let f (x) = 1/3x − 1. Sketch the graph of f, f , and y = x on the same set of axes. Solution We apply the four steps to graph the inverse function: Step 1. Sketch the graph of f . Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 16 Graphing Inverse Functions Graphing an Inverse Function Solution Continued 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. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 17 Graphing Inverse Functions Graphing an Inverse Function Solution Continued Step 4. Sketch the curve that contains the points plotted in step 3: The points from step 3 lie on a line. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 18 Finding an Equation of the Inverse of a Model Finding an Equation of the Inverse of a Model Example 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 2000. A reasonable model is f (t) = 0.73t + 1.54 −1 1. Find an equation of f . 2. Find f (10). What does it mean in this situation? Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 19 Finding an Equation of the Inverse of a Model Finding an Equation of the Inverse of a Model Example Continued 1. 2. 3. 4. −1 Find an equation of f . Find f (10). What does it mean in this situation? −1 Find f (10). What does it mean in this situation? What is the slope of f ? What does it mean in this situation? −1 5. What is the slope of f ? What does it mean in this situation? Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 20 Solution 1. Since f sends values of t to values of r, f values of r to values of t −1 −1 sends To find an equation of f , we want to write t in terms of r . Here are three steps to follow to −1 find an equation of f : Finding an Equation of the Inverse of a Model Finding an Equation of the Inverse of a Model Solution Exampl Continued Step 1. We replace f (t) with r : r = 0.73t + 1.54. Step 2. We solve the equation for t: -1 t = f (r) = 1.37r - 2.11 Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 22 Finding an Equation of the Inverse of a Model Finding an Equation of the Inverse of a Model Solution Exampl Continued −1 Step 3. Since f sends values of r to values of t, we −1 −1 have f (r) = t. So, we can substitute f (r ) for t in the equation t = 1.37r − 2.11: Check that the graph of f across the line y = x. −1 is the reflection of f Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 23 Finding an Equation of the Inverse of a Model Finding an Equation of the Inverse of a Model Solution Exampl Continued 2. f (10) = 0.73(10) + 1.54 = 8.84. 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 2010. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 24 Finding an Equation of the Inverse of a Model Finding an Equation of the Inverse of a Model Solution Exampl Continued −1 −1 3. f (10) = 1.37(10) − 2.11 = 11.59. Since f sends values of r to values of t, this means that t = 11.59 −1 when r = 10. According to the model f , the revenue will be $10 million in 2012. 4. The slope of f (t) = 0.73t + 1.54 is 0.73. This means that the rate of change of r with respect to t is 0.73. According to the model f , the revenue increases by $0.73 million each year. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 25 Finding an Equation of the Inverse of a Model Finding an Equation of the Inverse of a Model Solution Exampl Continued −1 5. The slope of f (r) = 1.37r −2.11 is 1.37. This means that the rate of change of t with respect to r −1 is 1.37. According to the model f , 1.37 years pass each time the revenue increases by $1 million. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 26 Finding an Equation of the Inverse of a Model Three-Step Process for Finding the Inverse Function Process Find the inverse of an invertible model f, where p = f (t). 1. Replace f (t) with p. 2. Solve for t. −1 3. Replace t with f (p). Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 27 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 Find the inverse of f (x) = 2x – 3. Solution Step 1. Substitute y for f (x): y = 2x − 3 Step 2. Solve for x: Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 28 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 Solution Example Continued −1 Step 3. Replace x with f (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: −1 To verify our work, check that the graph of f is the reflection of the graph of f across the line y = x. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 29 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 Solution Example Continued Process 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. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 30 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 Process Example Continued −1 3. Replace x with f (y). −1 4. Write the equation of f in terms of x. Definition 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. Copyright © 2011 Pearson Education, Inc. Section 11.1 Lehmann, Elementary and Intermediate Algebra, 1 ed Slide 31