Introduction to Relations

Domain and Range of a Relation

Any set of ordered pairs (x,y) is called a relation in x and y. Furthermore,

The set of first components in the ordered pairs is called the domain of the relation. The set of second components in the ordered pairs is called the range of the relation.

Definition of Function and Vertical Line Test

Definition of a Function

Given a relation in x and y, we say “y is a function of x” if for every element x in the domain, there corresponds exactly one element y in the range.

Examples

Determine if the relation is a function.

1. {(8,1),(9,10),(-6,14), (0,1)}

We are looking to make sure that every element in the domain (x) is only paired with one element in the domain.

8 is paired with only 1

9 is paired with only 10

-6 is paired with only 14

0 is paired with only 1

This relation is a function because for every element in the domain there corresponds exactly one element in the range.

2. {(6,3), (14,-3), (5,7),(14,1),(8,-12)}

This relation is NOT a function because the element 14 is paired with both -3 and

1.

Vertical Line Test

Consider a relation defined by a set of points (x,y) in a rectangular coordinate system. The graph defines y as a function of x if no vertical line intersects the graph in more than one point.

Examples

1. This relation is a function because it passes the vertical line test. The vertical line only intersects the graph at one point. It doesn’t matter where the vertical line is placed for this to be true.

2. This relation is NOT a function because the vertical line intersects the graph in more than one point.

Determining Whether an Equation Represents a Function

We have seen that not every set of ordered pairs defines a function. Similarly, not all equations with the variables x and y define a function. If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not