Section 5.2

One-to-One Functions;

Inverse Functions

OBJECTIVE 1

(a)

(b) {(1,5), (2,8), (3,11), (4,14)}

Dan

John

Joe

Andy

Saturn

Pontiac

Honda

Student Car

For each function, use the graph to determine whether the function is one-to-one.

(a) (b)

A function that is increasing on an interval I is a one-to-one function in I.

A function that is decreasing on an interval I is a one-to-one function on I.

OBJECTIVE 2

Dan

John

Joe

Michelle

Saturn

Pontiac

Honda

Chrysler

Student Car

Find the inverse of the following function. Let the domain of the function represent certain students, and let the range represent the make of that student’s car. State the domain and the range of the inverse function.

{(1,5), (2,8), (3,11), (4,14)}

State the domain and the range of the function and its inverse.

3 1 3

1

1 -1

Verify that the inverse of 2 is 2

by showing that for all in the domain of

and for all in the domain of .

g x x g x x

g g x x x g

g g x x x g

1

1

1

1 1 1Verify that the inverse of is .

2 1 2 2

For what values of is ?

For what values of is ?

f x f xx x

x f f x x

x f f x x

OBJECTIVE 3

x

y

(-1,0)

(0,3)

(1,4)

The graph shown is that of a one-to-one function. Draw the graph of the inverse.

OBJECTIVE 4

-1

1Find the inverse of 1.

3

Graph and on the same coordinate axes.

f x x

f f

2 1, 1

1

xf x x

x

Find the domain and the range of

2 1

1

xf x

x

2Find the inverse of 1 if 1.y x x