Intermediate Algebra (11th edition)

SECTION 10.1 Inverse Functions 573

For inverses ƒ and it follows that for all xin their domains,

and

Finding Inverses of One-to-One Functions

Decide whether each function is one-to-one. If it is, find the inverse.

(a)

Each x-value in Fcorresponds to just one y-value. However, the y-value 1 corre-

sponds to two x-values, and 0. Also, the y-value 2 corresponds to both 1 and 2.

Because some y-values correspond to more than one x-value, Fis not one-to-one and

does not have an inverse.

(b)

Every x-value in Gcorresponds to only one y-value, and every y-value corre-

sponds to only one x-value, so Gis a one-to-one function. The inverse function is

found by interchanging the x- and y-values in each ordered pair.

The domain and range of Gbecome the range and domain, respectively, of

(c) The table shows the number of days in which the air in Connecticut exceeded the

8-hour average ground-level ozone standard for the years 1997–2006.

G-^1.

G-^1 = 51 1, 3 2 , 1 2, 0 2 , 1 3, 2 2 , 1 0, 4 26

G= 51 3, 1 2 , 1 0, 2 2 , 1 2, 3 2 , 1 4, 0 26

- 2

F= 51 - 2, 12 , 1 - 1, 0 2 , 1 0, 12 , 1 1, 22 , 1 2, 226

EXAMPLE 1

1 ƒƒ^121 x 2 x 1 ƒ^1 ƒ 21 x 2 x.

ƒ-^1 ,

NOW TRY
EXERCISE 1
Decide whether each function
is one-to-one. If it is, find the
inverse.

(a)

(b)

(c)The number of stories and
height of several tall
buildings are given in the
table.

1 4, 2 2 , 1 9, 3 26

G= 51 0, 0 2 , 1 1, 1 2 ,

1 1, - 22 , 1 2, - 826

F= 51 - 1, - 22 , 1 0, 0 2

Stories Height

31 639

35 582

40 620

41 639

64 810

NOW TRY ANSWERS

1. (a)not one-to-one
   (b)one-to-one;

(c)not one-to-one

1 2, 4 2 , 1 3, 9 26

G-^1 = 51 0, 0 2 , 1 1, 1 2 ,

Let ƒ be the function defined in the table, with the years forming the domain and

the numbers of days exceeding the ozone standard forming the range. Then ƒ is not

one-to-one, because in two different years (2000 and 2006), the number of days with

unacceptable ozone levels was the same, 13. NOW TRY

Number of Days Number of Days

Year Exceeding Standard Year Exceeding Standard

1997 27 2002 36

1998 25 2003 14

1999 33 2004 6

2000 13 2005 20

2001 26 2006 13

Source:U.S. Environmental Protection Agency.

Horizontal Line Test

A function is one-to-one if every horizontal line intersects the graph of the func-

tion at most once.

OBJECTIVE 2 Use the horizontal line test to determine whether a func-

tion is one-to-one. By graphing a function and observing the graph, we can use the

horizontal line testto tell whether the function is one-to-one.

The horizontal line test follows from the definition of a one-to-one function. Any

two points that lie on the same horizontal line have the same y-coordinate. No two

ordered pairs that belong to a one-to-one function may have the same y-coordinate.

Therefore, no horizontal line will intersect the graph of a one-to-one function more

than once.