Exercise 3 Page 103

A relation is a set of $(x,y)$ coordinates. When each $x$ value corresponds with exactly one $y$ value, the relation is a function. Below we will provide examples of relations that are and are not functions.

A relation that is a function

Any set of ordered pairs in which each $x$ corresponds with exactly one $y$ is a function. Consider the following relation. Notice that each of the $x$ values $\text{-}1,3,2,1,$ and $5$ each correspond with only one $y$ value. Thus, the relation is a function. Consider another relation: Again, each $x$ value corresponds with exactly one $y$ value. Notice that the points $(1,4)$ and $(5,4)$ share the same $y$ value. The $y$ values do not affect if the relation is a function. In other words, it is okay for two $x$ values to share one $y$ value, as long as each $x$ value only correspond to one $y$ value. Thus, this relation is a function.

A relation that is not a function

A relation in which at least one $x$ value corresponds to more than one $y$ value is not a function. Consider the following relation.

Notice that the points $(\text{-}1,2)$ and $(\text{-}1,5)$ show that the $x$ value $\text{-}1$ corresponds to more than one $y$ value. Thus, the relation is not a function.