Functions
A function results when you discover a relationship between the values in a domain and the values in a range. Certain limitations apply to this relationship, however. First, each domain value must be unique. Second, each range value must correspond to only one domain value. Expressed differently, if you designate a number in the domain, then you find only one number in the range that corresponds to it. In this respect, since a one-to-one correspondence pertains between the values in the domain and the values in the range, the value of the number in the domain determines the value of the number in the range.
In Figure 4.10, a Cartesian system allows you to illustrate the functional relationship. This function establishes a pattern that relates the values of the domain with those of the range. The domain and range constitute sets. While the numbers in the domain form a union with the numbers in the range, it remains that the domain-range pairs that result are all unique. The equation that generates these pairs is y = x + 1.
Figure 4.10. A function describes the relation between the domain and the range in which a value in the domain determines a value in the range.
Given this equation, then, each value you designate for the domain generates a unique value. The equation, then, is a proper function. It is an equation or relationship that allows you to generate unique range values by using a set of domain values that are themselves unique. No two domain values generate the same range value.