chapter 5: exponential and logarithmic functions

Copyright © 2007 Pearson Education, Inc. Slide 5-2

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



Example

Also, f [g(12)] = 12. For these functions, it can be shown that

for any value of x. These functions are inverse functions of each other.

12)]12([i.e.129681

)96(

96128)12(

.81

)( and 8)(Let

fgg

f

xxgxxf

xxfgxxgf )]([and)]([


• Only functions that are one-to-one have inverses.

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).


5.1 One-to-One Functions

Example Decide whether the function is one-to-one.

(a) (b)

Solution

(a) For this function, two different x-values produce two different y-values.

(b) If we choose a = 3 and b = –3, then 3 –3, but

124)( xxf 225)( xxf

one.-to-one is),()( Since .124124

and 44 then , that Suppose

fbfafba

baba

one.-to-onenot is therefore),3()3( so ,4)3(25)3(and4325)3( 22

fffff


5.1 The Horizontal Line Test

Example Use the horizontal line test to determine whether the graphs are graphs of one-to-one functions.

(a) (b)

If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one.

Not one-to-one One-to-one



Exampleare inverse functions of each other.

Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if

. ofdomain in the every for ))((and

, ofdomain in the every for ))((

fxxxfg

gxxxgf

1)( and 1)( that Show 33 xxgxxf

xxxxfgxfg

xxxxgfxgf

3 33 3

33

11)]([))((

1111)]([))((


5.1 Finding an Equation for the Inverse Function

• Notation for the inverse function f -1 is read

“f-inverse”

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.


5.1 Example of Finding f -1(x)

Example Find the inverse, if it exists, of

Solution

.5

64)(

xxf

Write f (x) = y.5

64 xy

Interchange x and y.5

64 yx

Solve for y.

465645

x

y

yx

Replace y with f -1(x).4

65)(1 x

xf


5.1 The Graph of f -1(x)

• f and f -1(x) are inverse functions, and f (a) = b for

real numbers a and b. Then f -1(b) = a.

• 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.


5.1 Finding the Inverse of a Function with a Restricted Domain

Example Let

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.

The range of f is the domain of f -1, so its inverse is

).( Find.5)( 1 xfxxf

55

55

2

2

xyyx

yxxy

.0,5)( 21 xxxf


5.1 Important Facts About Inverses

1. If f is one-to-one, then f -1 exists.

2. The domain of f is the range of f -1, and the range of f is the domain of f -1.

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.


5.1 Application of Inverse Functions

Example Use the one-to-one function f (x) = 3x + 1 and the numerical values in the table to code the message BE VERY

CAREFUL.

A 1 F 6 K 11 P 16 U 21B 2 G 7 L 12 Q 17 V 22C 3 H 8 M 13 R 18 W 23D 4 I 9 N 14 S 19 X 24E 5 J 10 O 15 T 20 Y 25

Z 26

Solution BE VERY CAREFUL would be encoded as7 16 67 16 55 76 10 4 55 16 19 64 37

because B corresponds to 2, and f (2) = 3(2) + 1 = 7,and so on.

chapter 5: exponential and logarithmic functions

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