School Subjects
Math and Arithmetic
Algebra
Calculus

When can you say that a relation is a function?

## Answer

###### March 27, 2009 9:07PM

Relations and functions are very closely related. While all functions are relations, not all relations are functions. That's because functions are a special subset of relations. You can think of a relation as a set containing pairs of related numbers. For example, {(0,0), (1,1), (2,4), (3,9), (4,16)} represents a relation. There are five pairs of numbers. In each pair, the values of the second numbers (known as the range) are dependent upon the values of the first numbers (known as the domain). You can also think of the first number in each pair to be the x value and the second number to be the y value. In other words, y is dependent upon x. So, what makes a relation a function? For a relation to be a function, there must be one and only one y value for each x value. If there are two pairs of numbers that have the same x value but different y values, then the relation is NOT a function. In the above example, the domain is between zero and four, inclusive. Because each x value is unique and has only one corresponding y value, the relation is, in fact, a function. The function is y = x2, which can also be written f(x) = x2.