Let A and B be sets. A function from A to B is a nonempty relation f ⊆ A × B that satisfies the following two conditions: 1. Existence: For all a in A, there exists a b in B such that (a, b) ∈ f. 2. Uniqueness: If (a, b) ∈ f and (a, c) ∈ f , then b = c. That is, given any element a in A, there is one and only one element b in B such that (a, b) ∈ f. Set A is called the domain of f and is denoted by dom f. Set B is referred to as the codomain of f. We may write f : A → B to indicate f has domain A and codomain B. The range of f, denoted rng f, is the set of all second elements of members of f . That is,

 

 

If (x, y) is a member of f, we often say that f maps x onto y or that y is the image of x under f. It is also customary to write y = f (x) instead of (x, y) ∈ f. This agrees with the familiar usage when f is described by a formula, but also applies in more general settings. When a function consists of just a few ordered pairs, it can be identified simply by listing them. Usually, however, there are too many to list, so the function is identified by specifying the domain and giving a rule for determining the unique second element in the ordered pair that corresponds to any particular first element. When this rule is a formula, we are back to the intuitive notion of a function with which we began the section. Thus to say that a function f is given by the formula f (x) = x² + 3 means that

 

 

The domain of the function would be obtained either from the context or by stating it explicitly. Unless told otherwise, when a function is given by a formula, the domain is taken to be the largest subset of R for which the formula will always yield a real number. Having defined a function to be a set of ordered pairs satisfying the existence and uniqueness conditions, we should note that the notation f : A → B is slightly more restrictive than the ordered pair definition because it specifies a particular codomain. (Some authors refer to f : A → B as a mapping from A to B rather than a function from A to B, but it is common practice to treat the terms function and mapping as synonymous, as we shall do.) This subtle difference between the ordered pair definition and the f : A → B notation will be significant in Example 3.7, where we want to consider the functions

 

 

to be different functions (with different properties) even though their ordered pairs are identical.