Transcript Ch 5 Inverse, Exponential and Logarithmic Functions
Chapter 5: Exponential and Logarithmic Functions
5.1 Inverse Functions
5.2 Exponential Functions 5.3 Logarithms and Their Properties 5.4 Logarithmic Functions 5.5 Exponential and Logarithmic Equations and Inequalities 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions
Slide 5-2
Copyright © 2007 Pearson Education, Inc.
5.1 Inverse Functions
Example
Let
f
(
x
) 8
x
and
g
(
x
) 1 8
x
.
f
( 12 ) 8 12 96
g
( 96 ) 1 8 96 12 i.e.
g
[
f
( 12 )] 12 Also,
f
[
g
(12)] = 12. For these functions, it can be shown that
f
[
g
(
x
)]
x
and
g
[
f
(
x
)]
x
for any value of
x
. These functions are
inverse functions
of each other. Copyright © 2007 Pearson Education, Inc.
Slide 5-3
5.1 One-to-One Functions
A function
f
is a
one-to-one function
if, for elements
a
and
b
from the domain of
f
,
a
b
implies
f
(
a
)
f
(
b
).
• Only functions that are one-to-one have inverses.
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Slide 5-4
5.1 One-to-One Functions
Example
Decide whether the function is one-to-one.
(a)
f
(
x
) 4
x
12 (b)
f
(
x
) 25
x
2
Solution
(a) For this function, two different
x
-values produce two different
y
-values.
Suppose that
a
b
, then 4
a
4
a
12 4
b
12 .
Since
f
(
a
) 4
b
f
and (
b
),
f
is one to one.
(b) If we choose
a
= 3 and
b
= –3, then 3 –3, but
f
( 3 ) so
f
( 3 ) 25
f
( 3 2 3 ), 4 and therefore
f f
( 3 ) 25 is not one to ( 3 ) 2 one.
4 , Copyright © 2007 Pearson Education, Inc.
Slide 5-5
5.1 The Horizontal Line Test
If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one.
Example
Use the horizontal line test to determine whether the graphs are graphs of one-to-one functions.
(a) (b) Not one-to-one Copyright © 2007 Pearson Education, Inc.
One-to-one
Slide 5-6
5.1 Inverse Functions
Let
f
be a one-to-one function. Then,
g
is the
inverse function
(
f
g
)(
x
)
x
of
f
and
f
for every is the inverse of
g
if
x
in the domain of
g
, and (
g
f
)(
x
)
x
for every
x
in the domain of
f
.
Example
Show that
f
(
x
)
x
3 1 and
g
(
x
) 3 are inverse functions of each other.
(
f
g
)(
x
)
f
[
g
(
x
)] 3
x
1 3 1
x
1 1
x
1
x
(
g
f
)(
x
)
g
[
f
(
x
)] 3
x
3 1 1 3
x
3
x
Slide 5-7
Copyright © 2007 Pearson Education, Inc.
5.1 Finding an Equation for the Inverse Function
• Notation for the inverse function
f
“
f
-inverse” -1 is read
Finding the Equation of the Inverse of y = f(x) 1.
Interchange
x
and
y.
2.
Solve for
y.
3
. Replace
y
with
f
-1 (
x
).
Any restrictions on
x
and
y
should be considered.
Copyright © 2007 Pearson Education, Inc.
Slide 5-8
5.1 Example of Finding
f
-1
(
x
)
Example
f
(
x
) 4
x
Solution
5 Find the inverse, if it exists, of 6 .
y x
4
x
4
y
5 6 6 Write
f
(
x
) =
y
.
Interchange
x
and
y
.
5
x y
4 5
y x
5 6 6 Solve for
y
.
f
1 Copyright © 2007 Pearson Education, Inc.
(
x
) 5
x
4 6 4 Replace
y
with
f
-1 (
x
).
Slide 5-9
5.1 The Graph of
f
-1
(
x
)
•
f
and
f
-1 (
x
) are inverse functions, and
f
(
a
) =
b
real numbers
a
and
b
. Then
f
-1 (
b
) =
a
. for • If the point (
a
,
b
) is on the graph of
f
, then the point (
b
,
a
) is on the graph of
f
-1 .
If a function is one-to-one, the graph of its inverse
f
-1 (
x
) is a reflection of the graph of
f
across the line
y
=
x
.
Slide 5-10
Copyright © 2007 Pearson Education, Inc.
5.1 Finding the Inverse of a Function with a Restricted Domain
Example
Let
f
(
x
)
x
5 .
Find
f
1 (
x
).
Solution
Notice that the domain of
f
is restricted to [–5, ), and its range is [0, ). It is one-to-one and thus has an inverse.
x y y
2
x
x y
y x
2 5 5 5 5 The range of
f
is the domain of
f
-1 , so its inverse is
f
1 (
x
)
x
2 5 ,
x
0 .
Copyright © 2007 Pearson Education, Inc.
Slide 5-11
5.1 Important Facts About Inverses
1. If
f
is one-to-one, then
f
-1 exists.
2. The domain of range of
f f
is the range of
f
is the domain of
f
-1 .
-1 , and the 3. If the point (
a
,
b
) is on the graph of
f
, then the point (
b
,
a
) is on the graph of
f
-1 , so the graphs of
f
and
f
-1 are reflections of each other across the line
y
=
x
.
Slide 5-12
Copyright © 2007 Pearson Education, Inc.
5.1 Application of Inverse Functions
Example
Use the one-to-one function
f
(
x
) = 3
x
+ 1 and the numerical values in the table to code the message BE VERY CAREFUL.
A 1 B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 K 11 L 12 M 13 N 14 O 15 P 16 Q 17 R 18 S 19 T 20 U 21 V 22 W 23 X 24 Y 25 Z 26
Solution
and so on.
BE VERY CAREFUL would be encoded as 7 16 67 16 55 76 10 4 55 16 19 64 37 because B corresponds to 2, and
f
(2) = 3(2) + 1 = 7, Copyright © 2007 Pearson Education, Inc.
Slide 5-13