Intermediate Algebra (11th edition)

(Marvins-Underground-K-12) #1

572 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions


OBJECTIVES In this chapter we study two important types of functions, exponential and


logarithmic. These functions are related: They are inversesof one another.


OBJECTIVE 1 Decide whether a function is one-to-one and, if it is, find its


inverse.Suppose we define the function


We can form another set of ordered pairs from Gby interchanging the x- and y-values


of each pair in G. We can call this set F, so


To show that these two sets are related as just described, Fis called the inverseof G.


For a function ƒ to have an inverse, ƒ must be a one-to-one function.


F= 512 , - 22 , 11 , - 12 , 10 , 02 , 13 , 12 , 15 , 226.


G= 51 - 2 , 22 , 1 - 1 , 12 , 10 , 02 , 11 , 32 , 12 , 526.


Inverse Functions


10.1


1 Decide whether a
function is one-to-
one and, if it is, find
its inverse.
2 Use the horizontal
line test to
determine whether
a function is one-to-
one.
3 Find the equation
of the inverse of a
function.
4 Graph given the
graph of ƒ.

ƒ-^1
One-to-One Function

In a one-to-one function,each x-value corresponds to only one y-value, and each


y-value corresponds to only one x-value.


The function shown in FIGURE 1(a)is not one-to-one because the y-value 7 corre-


sponds to two x-values, 2 and 3. That is, the ordered pairs and both


belong to the function. The function in FIGURE 1(b)is one-to-one.


1 2, 7 2 1 3, 7 2


The inverseof any one-to-one function ƒ is found by interchanging the com-


ponents of the ordered pairs of ƒ. The inverse of ƒ is written Read as “the


inverse of ƒ ”or “ƒ-inverse.”


ƒ^1. ƒ-^1


CAUTION The symbol does notrepresent.


1


ƒ 1 x 2


ƒ-^11 x 2


The definition of the inverse of a function follows.


Inverse of a Function

The inverseof a one-to-one function ƒ, written , is the set of all ordered pairs


of the form where belongs to ƒ. Since the inverse is formed by


interchanging xand y, the domain of ƒ becomes the range of and the range


of ƒ becomes the domain of ƒ-^1.


ƒ-^1


1 y, x 2 , 1 x, y 2


ƒ-^1


Domain
1
2
3
4
5

6
7
8
9

Range

Not One-to-One

1
6

Domain Range

7
8

5
2
3
4

One-to-One
(a) (b)
FIGURE 1
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