A relation is a set of $(x, y)$ coordinates. When each $x$ value corresponds with exactly one $y$ value, the relation is a function. Below we will provide examples of relations that are and are not functions.

A relation that is a function

Any set of ordered pairs in which each

x

corresponds with exactly one

y

is a function. Consider the following relation.

(- 1, 2), (3, 5), (2, - 3), (1, 4), (5, - 1)

Notice that each of the

x

values

- 1, 3, 2, 1,

and

5

each correspond with only one

y

value. Thus, the relation is a function. Consider another relation:

(- 1, 2), (3, 5), (2, - 3), (1, 4), (5, 4)

Again, each

x

value corresponds with exactly one

y

value. Notice that the points

(1, 4)

and

(5, 4)

share the same

y

value. The

y

values do not affect if the relation is a function. In other words, it is okay for two

x

values to share one

y

value, as long as each

x

value only correspond to one

y

value. Thus, this relation is a function.

A relation that is not a function

A relation in which at least one $x$ value corresponds to more than one $y$ value is not a function. Consider the following relation.

(- 1, 2), (- 1, 5), (2, - 3), (1, 4), (5, - 1)

Notice that the points $(- 1, 2)$ and $(- 1, 5)$ show that the $x$ value $- 1$ corresponds to more than one $y$ value. Thus, the relation is not a function.

Exercise