inverses & one-to-one

Copyright © 2011 Pearson Education, Inc. Slide -1

CHAPTER 6

INVERSES & ONE-to-ONE Functions


Inverse 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.

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).


One-to-One Functions

Example Decide whether each 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


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


Inverse Functions

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)]([))((


Finding an Equation for the Inverse Function

.

Finding the Equation of the Inverse of y = f(x)

For a one-to-one function f defined by an equation

y = f(x), find the defining equation of the inverse as follows. (Any restrictions on x and y should be considered.)

1. Interchange x and y.

2. Solve for y.

3. Replace y with f -1(x).


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


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.


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


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.


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.