Inverse Functions

Consider the function f illustrated by the mapping diagram.

The function f takes the domain values of 1, 8 and 64 and produces the corresponding range values of 6, 7, and 9.

1 6

8 7

64 9

f

Inverse Functions

Now consider the function g illustrated in the mapping diagram.

The function g "undoes" function f.

1 6

8 7

64 9

f

g

It takes the f (x) range values of 6, 7, and 9 as its domain values and produces as its range values, 1, 8, and 64 which were the domain values of f (x).

Inverse Functions

The mapping diagram with the domains and ranges of f (x) and g(x) are labeled is shown.

g

1 6

8 7

64 9

f

g

domain of f (x) range of f (x)

range of g(x) domain of g(x)

Inverse Functions

If there exists a one-to-one function, g(x), that "undoes" f (x) for every value in the domain of f (x), then g(x) is called the inverse function of f (x) and is denoted f - 1(x).

g

1 6

8 7

64 9

f

g

domain of f (x) range of f (x)

range of g(x) domain of g(x)

Inverse Functions

DEFINITION:

Let f and g be functions where

f (g (x )) = x for every x in the domain of g

and

g ( f (x )) = x for every x in the domain of f.

Then function g is the inverse of function f, and is denoted

f - 1 (x )

To see why the definition is written this way, consider g(f (x)) = x.

Inverse Functions

The part that is done first is inside parentheses.

The function g then takes this range value of the f function, f (x), as its domain value and produces x (the original domain value of the f function) as its range value.

This means the function f takes as its domain value, x, and produces the range value, f (x).

Inverse Functions

x f (x)

domain value of function f, x

range value of function f, f (x)

range value of function g, x

domain value of function g, f (x)

g

f

Inverse Functions

Example: Algebraically show that the one-to-one functions, and g(x) = (x – 5)3, are inverses of each

other. ,53 xxf

First, show that (f g)(x) = x.

Next, show that (g f)(x) = x.

(f g)(x) = 553 3 x 55 x = x.

(g f)(x) = 33 55 x 33 x = x.

Inverse Functions

Try: Algebraically show that the one-to-one functions,

,8

3 xxgf (x) = 8x + 3, and are inverses of each

other.

(f g)(x) = 38

38

x

= x – 3 + 3 = x.

(g f)(x) =

.8

88

338x

xx

Inverse Functions

A PROPERTY OF INVERSE FUNCTIONS

x f (x)

domain of f range of f

range of f - 1 domain of f - 1f - 1

f

The range of a function, f, is the domain of its inverse, f - 1.

The domain of a function, f, is the range of its inverse, f - 1.

Inverse Functions

ANOTHER PROPERTY OF INVERSE FUNCTIONS

The graphs of a function, f, and its inverse, f - 1, are symmetric across the line y = x.

For example, the graphs of and f - 1(x) = x3 3 xxf

are shown along with the graph of y = x.

1

- 1

- 2 2

Inverse Functions