Function

In order to define a function, we need to know about mathematical "relations". Simply put, a relation is a relationship between two sets of information. For instance, an ordered pair like (1, 10) is a relation, where 1 is the x-value on a graph and 10 is the y-value on a graph. The ordered pair (1, 10) is a position on a graph that is 1 unit to the right and 10 units up from (0, 0) or the origin. Example 1 shows how relations can be represented.

Example 1: Three students are competing in a challenge. First place gets 10 tokens, second place gets 7 tokens, and third place gets 5 tokens. The relationship between place and tokens is a relation, and we commonly represent it using ordered pairs {(1, 10), (2, 7), (3, 5)}. Below are three additional ways to represent relations.

So, what is a function? A function is a special type of relation that associates one quantity with only one other quantity. Therefore, the relation described in Example 1 is also a function because each quantity in the "Place" category is associated with only one quantity in the "Token" category. In other words, there is only one result for coming in first place (10 tokens), only one result for second place (7 tokens), and only one result for third place (5 tokens).