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Inverse Functions

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?
f = {(3, 3), (5, 9), (6, 3)}
g = {(1, 2), (2, 3), (4, 5)}
h = {(3, 7), (4, 4), (7, 3)}

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 f-1

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 :
(f o g)(x) = x for all x in dom g
(g o f)(x) = x for all x in dom f

In other words, the machines f o g and g o f 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.

Solution

Since this cannot be simplified into x , we may stop and conclude that f and g are not inverses.

Even though the first one worked, they both have to work. So we conclude that f and g are not
inverses of each other.

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

Solution

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

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)

Notice that f is its own inverse.

b)

Basic Inverses Examples

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 D are inverses of each other.
 E is 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.


where k is the function graphed to the right.

Solution

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