Relations and Functions

When we want to use mathematics to model the real world, we frequently want to look at the relationship between two entities, which we represent with two variables. For each instance, we have associated values for the two variables, which we represent using ordered pairs of the form $\left(x,y\right).$

In general, if the value of the second component, y, depends on the value of the first component, x, then y is the dependent variable, and x is the independent variable.

Any set of one or more ordered pairs is called a relation. A function is a special case of a relation.

Definition of a Function

A relation is a set of ordered pairs. A function is a relation for which, for each distinct value of the first component of the ordered pairs, there is exactly one value of the second component.

Example

Determine whether each relation defines a function.

$F=\lbrace\left(3,4\right),\left(-4,1\right),\left(2,7\right)\rbrace$

$G=\lbrace\left(3,2\right),\left(3,3\right),\left(3,7\right),\left(4,5\right)\rbrace$

$H=\lbrace\left(2,2\right),\left(3,2\right),\left(4,2\right),\left(5,-1\right)\rbrace$

Solution

$F\:$is a function because for each different x-value there is exactly one y-value. 

3 maps to 4

-4 maps to 1

2 maps to 7

$G\:$is NOT a function because one x-value maps to more than one y-value. Namely,

3 maps to 2, 3, and 7

4 maps to 5

$H\:$is a function because although several of the x-values map to the same y-value, each of the x-values only maps to exactly one y-value.

2 corresponds to 2

3 corresponds to 2

4 corresponds to 2

5 corresponds to -1