# Inverse Functions

**Due Homework**

8 webwork problems (1 hint)

**Undue Homework**

Textbook 2.5: 7, 9, 53, 55

Overview

• Invertability.

• Definition of an Inverse Function.

• Expressions and Inverses .

• Basic Inverses Examples.

• Graphs and Inverses .

• The Horizontal Line Test .

• Graphin an Inverse.

• Machines and Inverses.

**Invertability**

In section 2.1, we determined whether a relation was a function by looking for
duplicate x- values .

Invertability is the opposite. A function is invertible if and only if it
contains no two ordered

pairs with the same y-values, but different x -values. Thus, to determine if a

function is invertible, we look for duplicate y-values. Invertible functions are
also called one-to-one.

Example

Which functions are invertible?

Solution

f is not invertible since it contains both (3, 3) and (6, 3).

g is invertible.

h is invertible.

**Definition of an Inverse Function**

If f is an invertible function, its inverse, denoted f^{–1}

, is the set of ordered pairs (y, x) such

that (x, y) is in f. That is, f^{–1} is f with its x- and y-values swapped. f^{–1}(x)
is not

1/f(x).

Example

Find the inverses of the invertible functions from the last example.

Solution

g–1 = {(2, 1), (3, 2), (5, 4)}

h–1 = {(7, 3), (4, 4), (3, 7)}

Note

1. Invertability insures that the a function’s inverse is a function.

2. A function can be its own inverse. Observe how the function h in the last
example has this

property .

3. Whenever g is f’s inverse then f is g’s inverse also.

4. Inversion swaps domain with range. That is

dom f = ran f^{-1}

ran f = dom

**Expressions and Inverses**

Example

Describe in words what the function f(x) = x does to its input.

Solution

Nothing.

The Cancellation Theorem

Functions f are g are inverses of each other if and only if both of the
following cancellation laws

hold:

()(x) = x for all x in dom g

()(x) = x for all x in dom f

In other words, the machines and
do nothing to their inputs. This means that
f reverses

all changes made by g and vise versa. In essence, f and g cancel each other out.

Example

Verify that the following pairs are inverses of each other.

a)

b)

c)

Solution

a)

b)

**Example**

If f(–7) = 8, and f is invertible, solve 1/2 f (x – 9) = 4.

Solution

1/2 f(x – 9) = 4

f(x – 9) = 8

f^{-1}(f(x – 9)) = f^{-1}(8)

x – 9 = –7

x = 2

Let f and g be inverses of each other, and let f(x) = y. Then by the
Cancellation

Theorem

g(y) = g(f(x)) = x

This partly proves the next theorem.

Change of Form Theorem

Functions f and g are inverses of each other if and only if both of the
following change of form

laws holds :

f(x) = y implies g(y) = x

g(x) = y implies f(y) = x

Change of Form Theorem (alternate version)

If f is invertible then

f(x) = y if and only if f^{–1}(y) = x

Example

If f(4) = 3, f(3) = 2, and f is invertible, find f^{–1}(3) and (f(3))^{–1}.

Solution

f^{–1}(3) = 4

(f(3))^{–1} = 2^{–1} = 1/2

To find the inverse of a function, f, algebraically

1. Set y = f(x).

2. Swap x with y.

3. Solve for y.

4. Replace y with f^{-1}(x).

**Example**

Find the inverses of

Solution

a)

y = 5 – x

x = 5 – y

x + y = 5

y = 5 – x

f^{–1}(x) = 5 – x

Notice that f is its own inverse.

b)

**Basic Inverses Examples**

f(x) | |||||||

f^{–1}(x) |
|||||||

Applies When |

**Graphs and Inverses**

To find f^{–1}(a) from the graph of f, start by finding a on the y-axis
and move horizontally until

you hit the graph. The answer is the x- value of the point you hit.

Example

Use the graph of f to find f^{–1}(2) and f^{–1}(3).

Solution

f^{–1}(2) = 3

f^{–1}(3) = 3.6

**The Horizontal Line Test**

The graph of a function is that of an invertible function if and only if every
horizontal line

passes through no or exactly one point.

Example

Which graph is that of an invertible function?

Solution

B, C, D, and E

Graphing an Inverse

To graph f^{–1} given the graph of f, we place a point (b, a) on the
graph of f^{–1} for

every point (a, b) on the graph of f. This has the effect of reflecting the

graph of f across the line y = x.

Example

a) Which pair of functions in the last example are inverses of each other?

b) Which function is its own inverse?

c) Which function is invertible but its inverse is not one of those shown?

Solution

B and Dare inverses of each other.

Eis its own inverse.

C is invertible, but its inverse is not shown

To graph f^{–1} given the graph of f, do the
following

1. Label several points (a, b) on f that

define its general shape.

2. For each, plot (b, a).

3. Draw the line y = x.

4. Connect the dots paying attention to the way

the graph is being reflected across y = x.

Example

Graph the inverse of the function, k graphed to

the right.

Machines and Inverses

From a machine perspective, a function f is invertible if and only if it is a
composition of

invertible operations (CIO). In this case, f^{-1} is the machine that performs the
opposite

operations in the opposite order (4O). When a function is a CIO, the machine
metaphor

is a quick and easy way to find its inverse. I will teach you how to do it using
a machine table, and

I may require you to show a machine table because otherwise there is no work to
show. However, that is the

point. With some practice, you can use this method to find inverses in your
head.

Example

Make a machine table for each function. If it is invertible find its inverse
using the machine table.

a) f(x) =

b) g(x) =

c) h(x) = where k is the function graphed to
the right.

Solution

a)

Prev | Next |