One-to-One Functions (Section 3.7, pp ) and Their Inverses

ACTIVITY 29:

One-to-One Functions (Section 3.7, pp. 280-285)and Their Inverses

Definition of a One-One Function:

A function with domain A is called a one-to-one function if no two elements of A have the same image, that is,

f(x1) ≠ f(x2) whenever x1 ≠ x2.An equivalent way of writing the above condition is:

If f(x1) = f(x2), then x1 = x2.

Horizontal Line Test:

A function is one-to-one if and only if no horizontal line intersects its graph more than once.

Example 1:

Show that the function f(x) = 5 − 2x is one-to-one.

This line clearly passes the horizontal line test. Consequently, it is a one to one functions.

Example 2:

Graph the function f(x) = (x−2)2 −3. The function is not one-to-one: Why? Can you restrict its domain so that the resulting function is one-to-one?

The function is not one to one because it does not pass the horizontal line test.

Restricting the domain to all

x> 2 makes the function one to one.

Definition of the Inverse of a Function:Let f be a one-to-one function with domain A and range B. Then its inverse function f−1 has domain B and range A and is defined by

f−1(y) = x if and only if f(x) = y,for any y B.∈

Example 3:

Suppose f(x) is a one-to-one function.If f(2) = 7, f(3) = −1, f(5) = 18, f−1 (2) = 6 find:

f−1(7) =

f(6) =

f−1(−1) =

f(f−1(18)) =

2

2

3

f(5) = 18

If g(x) = 9 − 3x, then g−1(3) =xg )3(Let 1 )( then xg 3

339ly Consequent x9 9

63 x

36

x 2

Property of Inverse Functions:

Let f(x) be a one-to-one function with domain A and range B.The inverse function f−1(x) satisfies the following “cancellation” properties:

1.f−1(f(x)) = x for every x A ∈2.f(f−1(x)) = x for every x B∈

Conversely, any function f−1(x) satisfying the above conditions is the inverse of f(x).

Example 4:

Show that the functions f(x) = x5 and g(x) = x1/5 are inverses of each other.

))(( xgf

5

1xf

55

1

x x

))(( xfg )( 5xg 515x x

Example 5:

Show that the functions

are inverses of each other.

xxxf

2531)(

3215)(

xxxg

)(xgf

3215

xxf

321525321531

xxxx

321525321531

xxxx

32152

32325

32153

32321

xx

xx

xx

xx

32

15232532

15332

xxx

xxx

32

15232532

15332

xxx

xxx

322101510

3231532

xxx

xxx

3217

3217

x

xx

1732

3217

xxx x

How to find the Inverse of a One-to-One Function:1. Write y = f(x).2. Interchange x and y. 3. Solve this equation for x in terms of y (if possible). The

resulting equation is y = f−1(x).

Example 6:

Find the inverse of f(x) = 4x−7.

74 xy74 yxyx 47

yx

4

7

47)(1

xxf

Example 7:

Find the inverse of

21

x

y

21)(

x

xf

21

y

x

)2(2

12

yy

yx

12 yx12 xxy

xxy 21

xxy 21

xxxf 21)(1

Example 8:

Find the inverse of

22)(

xxxf

22

xxy

22

yyx

22

22

yyyyx

yyx 22

yxxy 22xyxy 22

xxy 221

122

x

xy 122)(1

xxxf

Graph of the Inverse Function:

The principle of interchanging x and y to find the inverse function also gives us a method for obtaining the graph of f−1 from the graph of f. The graph of f−1 is obtained by reflecting the graph of f in the line y = x.The picture on the right hand side shows the graphs of:

4)( xxf 0 ,4)( and 21 xxxf

Example 9:

Find the inverse of the function

xxf 11)(

Find the domain and range of f and f−1. Graph f and f−1 on the same Cartesian plane.

xy 11

yx 11

yx 11

22 11 yx

yx 11 2

yx 11 2

11)( 21 xxf

xxf 11)( 11)( 21 xxf

Domain for f(x)1x

Range for f(x)1y

Range of f-1(x)

1y

Domain of f-1(x)

1x