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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