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