In this chapter we explore the concept of a mathematical function. Undoubtedly, functions have been prevalent in your previous mathematical experiences. You may have spent a lot of time working with functions given by an explicit formula, such as. In this chapter we develop a broader understanding of functions, presenting a commonly used and intuitive (but incomplete) definition in Section 5.1, followed by a variety of related terminology and concepts, and then a rigorous definition in Section 5.5.

What is a Function?

The notion of a function is broader than numerical functions that are given by explicit formulas.

DEFINITION 5.1 (Initial Version). Let A and B be sets. A function from A to B is a rule that assigns to each unique associated element.

There are two standard ways to represent “f is a function from A to B.” One is to write

and the other is

Both notations are supposed to indicate that the function f takes an input element and outputs an element, and we express this by writing f (a) = b. In the sections that follow, you will often read statements like “Let f : A → B.” This is a short but complete sentence stating that f is a function from A to B.

For example, consider the function g defined in a piecewise fashion for real numbers x:

The formulas and both give real numbers if is a real number, so we could use the notation g : because both A and B equal. As another example, you might have encountered parametric curves which trace out patterns in the plane. One of these gives a circle parametrized by

as illustrated in Figure 1. Here the input is a real number t ∈ R and the output is a point, so we write or.