Domain

The domain of a relation is the set that contains all the first elements, x, from the ordered pairs (x,y) that make up the relation. In mathematics, a relation is defined as a set of ordered pairs (x,y) for which each y depends on x in a predetermined way. If x represents an element from the set X, and y represents an element from the set Y, the Cartesian product of X and Y is the set of all possible ordered pairs (x,y) that can be formed with an element of X being first. A relation between the sets X and Y is a subset of their Cartesian product, so the domain of the relation is a subset of the set X. For example, suppose that X is the set of all men and Y is the set of all women. The Cartesian product of X and Y is the set of all ordered pairs having a man first and women second. One of the many possible relations between these two sets is the set of all ordered pairs (x,y) such that x and y are married. The set of all married men is the domain of this relation, and is a subset of X. The set of all second elements from the ordered pairs of a relation is called the range of the relation, so the set of all married women is the range of this relation, and is a subset of Y. The variable associated with the domain of the relation is called the independent variable. The variable associated with the range of a relation is called the dependent variable.

Many important relations in science, engineering, business and economics can be expressed as functions of real numbers. A function is a special type of relation in which none of the ordered pairs share the same first element. A real-valued function is a function between two sets X and Y, both of which correspond to the set of real numbers. The Cartesian product of these two sets is the familiar Cartesian coordinate system, with the set X associated with the x-axis and the set Y associated with the y-axis. The graph of a real-valued function consists of the set of points in the plane that are contained in the function, and thus represents a subset of the Cartesian plane. The x-axis, or some portion of it, corresponds to the domain of the function. Since, by definition, every set is a subset of itself, the domain of a function may correspond to the entire x-axis. In other cases the domain is limited to a portion of the x-axis, either explicitly or implicitly.

Example 1. Let X and Y equal the set of real numbers. Let the function, f, be defined by the equation y= 3x² + 2. Then the variable x may range over the entire set of real numbers. That is, the domain of f is given by the set D = {x| - ∞ ≤ x ≥ ∞}, read "D equals the set of all x such that negative infinity is less than or equal to x and x is less than or equal to infinity."

Example 2. Let X and Y equal the set of real numbers. Let the function f represent the location of a falling body during the second 5 seconds of descent. Then, letting t represent time, the location of the body, at any time between 5 and 10 seconds after descent begins, is given by f(t) = ¹⁄₂gt². In this example, the domain is explicitly limited to values of t between 5 and 10, that is, D = {t| 5 ≤ t ≥ 5}.

Example 3. Let X and Y equal the set of real numbers. Consider the function defined by y = PIx², where y is the area of a circle and x is its radius. Since the radius of a circle cannot be negative, the domain, D, of this function is the set of all real numbers greater than or equal to zero, D = {x| x ≥ 0}. In this example, the domain is limited implicitly by the physical circumstances.

Example 4. Let X and Y equal the set of real numbers. Consider the function given by y = 1/x. The variable x can take on any real number value but zero, because division by zero is undefined. Hence the domain of this function is the set D = {x| x NSIME 0}. Variations of this function exist, in which values of x other than zero make the denominator zero. The function defined by y = ¹⁄₂-x is an example; x=2 makes the denominator zero. In these examples the domain is again limited implicitly.