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# Relations and Functions by Christopher Monahan

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 domain, and the set of second elements (output values) is the range. The domain of 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 DB = {0, 5, 4, 2}, and the range of B is RB = {-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.