Definition (Mapping, Domain, Range). A mapping or relation between 2 sets, and is a rule assigning some elements from (called the domain) to an element in (called the range).

Example 1. If the set were the set of positive real numbers and the set were the set of negative real numbers, the rule “take the negative of an element” assigns each positive integer in to an element in .

Mathematically, we say that is a relation from a domain to a range using the following shorthand.

The rule of the function is often written as where . An element in the range can thus be given by . In the example above, .

Definition (Function). A mapping is defined as a function if it assigns each element in its domain to exactly one element in its range . The domain and range of are sometimes referred to as and respectively.

Example 2. The mapping where (example 1) is a function.

Example 3. (Counter-example) The mapping where is not a function since or .

It is common to write a function in the following manner instead.

,

Example 4. The function in example 1 can be written as ,

We can use a vertical line test on the graph of to determine whether is the rule of a function or not.

is a function if and only if every vertical line where intersects the graph of exactly once.