Functions

A relation is a function if no two of the ordered pairs in the relation have the same first coordinate and different second coordinates. Some, but not all, relations are functions. R₁ is not a function because two of its ordered pairs, (1, 1) and (1, 0), have the same first coordinates but different second coordinates. Similarly, R₇ and R₈ are not functions: (1, 1) and (1, –1) are both in R₇ and (3, 4) and (3, –4) are both in R₈. All of the other relations in the list are functions.

EXAMPLE 1: If a and b are real numbers, and if {(1, 2), (3, 4), (a, 6), (7, b)} is a function, what do we know about a and b?

Since a function cannot have two different ordered pairs with the same first coordinate, a cannot be 1 or 3 and a can be 7 only if b is 6. Other than that, there are no restrictions on a. If a is 7, b must be 6; if a is not 7, then b can be any number whatsoever.

If you look at the graphs of R_1, R₇, and R₈, the relations that are not functions, you can see that in each of them it is possible to draw at least one vertical line that intersects the graph at more than one point.

Key Fact N1