Inverse

Inverse of a Relation

The Inverse of a Relation Pictorially

If there is a relation between two sets A and B, interchanging the sets A and B gives the inverse of the relation.

If a relation is denoted by R, then its inverse is written as R⁻¹.

If the relation is shown by an arrow diagram, reversing the direction of the arrows gives its inverse.

If the relation is given as a set of ordered pairs (x, y), then interchanging the variables gives the inverse of the relation, i.e., the inverse is (y, x).

Therefore:

• Domain of R = Range of R⁻¹
• Range of R = Domain of R⁻¹

Example 1

Let A = {2, 3, 5, 6} and B = {4, 6, 10}. The relation "is a factor of" is shown pictorially.

The inverse of this relation is "is a multiple of".

Example 2

Find the inverse of the relation R = { (x, y) : x + 3 ≥ y }

Solution

To find R⁻¹, interchange x and y: x + 3 ≥ y becomes y + 3 ≥ x

So, R⁻¹ = { (x, y) : y + 3 ≥ x }

Example 3

Find the inverse of the relation R = { (x, y) : y = 2x }

Solution

Interchange x and y in the equation y = 2x: x = 2y → y = ½x

So, R⁻¹ = { (x, y) : y = ½x }

A Graph of the Inverse of a Relation

To determine whether a function has an inverse that is also a function, use the horizontal line test.

If any horizontal line intersects the graph of a function in only one point, then the function has an inverse that is also a function.

For example: The function y = 3x + 2 passes the horizontal line test. Therefore, it has an inverse that is also a function.

y = 3x + 2

passes the horizontal line test. Therefore, it has an inverse that is also a function.