copyright © 2011 pearson education, inc. logarithmic functions chapter 11

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 edSection 11.1 Slide 3Copyright © 2011 Pearson Education, Inc.

Definition of an Inverse of a FunctionFahrenheit/Celsius Relationship

Observations



Observations Continued



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.




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



Definition of an Inverse of a FunctionEvaluating 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


Definition of an Inverse of a FunctionEvaluating f and f

−1

1. f (3) = 27.

2. 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



−1

The graph of an invertible function f is shown.

1. Find f (2).

2. Find f −1(5).

Warning Continued

Example



−1

1. The blue arrows in the figure show that f sends 2 to 3. So, f (2) = 3.

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


Definition of an Inverse of a FunctionFinding 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 1

16 .2

x

f x

Solution

( )

0 16

1 8

2 4

3 2

4 1

x f x


Definition of an Inverse of a FunctionFinding 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


Graphing Inverse FunctionsComparing the graphs of a Function and Its Inverse

Sketch the graphs of f (x) = 2x , f −1, and y = x on the same set of axes.

Solution

Example


a


ExampleSolution Continued


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).


Property

Process


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 FunctionsGraphing an Inverse Function

Process Continued

Example

Solution


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.


Solution Continued


Step 4. Sketch the curve that contains the points plotted in step 3: The points from step 3 lie on a line.


Solution Continued


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. 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 ModelFinding an Equation of the Inverse of a Model

Example


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


ExamplExample Continued

Solution


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 + 1.54.

Step 2. We solve the equation for t:


ExamplSolution Continued


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.




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.




3. f −1(10) = 1.37(10) − 2.11 = 11.59. Since f −1 sends values of r to values of t, this means that t = 11.59 when r = 10. According to the model f −1, 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.




5. The slope of f −1 (r) = 1.37r −2.11 is 1.37. This means that the rate of change of t with respect to r is 1.37. According to the model f −1, 1.37 years pass each time the revenue increases by $1 million.




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 ModelThree-Step Process for Finding the Inverse Function

Process


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


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.





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.




Process


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.



ExampleProcess Continued

Definition

copyright © 2011 pearson education, inc. logarithmic functions chapter 11

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