# Relations and Functions

A ** relation** is any set of ordered pairs.

*A*= {(3, 4), (2, −1), (7, 2), (2, 0)} is a relation (not a very interesting one, but a relation). The set of first elements (input values) in the relation is called the

**, and the set of second elements (output values) is the**

*domain***. The domain of**

*range**A*is {2, 3, 7} and the range is {-1, 0, 2, 4}. The number 2 is used twice as the input value for

*A*, but it only has to be mentioned once in the domain statement.

**Example:** What are the domain and range of the relation *B*= {(0,-3), (5, 2), (4, −3), (2, −1), (9, 3)}?

The domain is the set of all first elements, so *D**B* = {0, 5, 4, 2}, and the range of *B* is *R**B* = {-3, 2, −1}. *D* and *R* represent the domain and range, and the subscript identifies the relation in question.

A graphical test for whether a graph represents a function is the vertical-line test. If a vertical line can cross a graph more than once, the graph does not represent a function.

A special type of relation is a function. A ** function** is a relation in which each element in the domain is paired with a unique element in the range. Of the two previous examples,

*A*is not a function because the input value 2 is associated with the output values of −1 and 0.

*B*is a function because each input value has a unique output. You may have noticed that −3 is used twice as an output in

*B*, and that is acceptable. The requirement for a function is that the output for each input must be unique, not the other way around.