Functions Given two sets 𝐴 and 𝐵, a set with elements that are ordered pairs (𝑥,𝑦), where 𝑥 is an element of 𝐴 and 𝑦 is an element of 𝐵, is a relation from 𝐴 to 𝐵. A relation from 𝐴 to 𝐵 defines a relationship between those two sets. A function is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the input; the element of the second set is called the output. Functions are used all the time in mathematics to describe relationships between two sets. For any function, when we know the input, the output is determined, so we say that the output is a function of the input. For example, the area of a square is determined by its side length, so we say that the area (the output) is a function of its side length (the input). The velocity of a ball thrown in the air can be described as a function of the amount of time the ball is in the air. The cost of mailing a package is a function of the weight of the package. Since functions have so many uses, it is important to have precise definitions and terminology to study them.

We can also visualize a function by plotting points (𝑥,𝑦) in the coordinate plane where 𝑦=𝑓(𝑥). The graph of a function is the set of all these points. For example, consider the function 𝑓, where the domain is the set 𝐷={1,2,3} and the rule is 𝑓(𝑥)=3−𝑥. In Figure 1.5, we plot a graph of this function.