Technical Mathematics, Sixth Edition

4.1. Functions and Relations

We noted in the introduction that a function or relation is a way of relating two quantities, such as relating the area of a circle to its radius. We will soon see that a function or relation can take several forms (graphs and tables, for example) but we will usually work in the form of an equation. So for now let's define relations and functions in terms of equations, and expand our definition later.

An equation that enables us to find a value of y for a given value of x is called a relation.

Example 1: The equation y = 3x − 5 is a relation. For any value of x, say x =2, there is a value of y (in this case, y = 1).

An equation that enables us to find exactly one value of y for a given value of x is called a function. Thus while the equation of Example 1 is a relation, it is also a function, because it gives just one value of y for any value of x.

Thus a function may be in the form of an equation. But not every equation is a function.

Example 2: The equation is not a function. Each value of x yields two values of y. This equation is a relation.

We see from Examples 1 and 2 that a relation need not have one value of y for each value of x, as is required for a function. Thus only some equations that are relations can also be called functions.

The fact that a relation is not a function does not imply second-class status. Relations are no less useful ...