Domain of a Function

From Department of Mathematics at UTSA
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Definition

In mathematics, the domain or set of departure of a function is the set into which all of the input of the function is constrained to fall. It is the set X in the notation f: X → Y, and is alternatively denoted as dom(f). Since a (total) function is defined on its entire domain, its domain coincides with its domain of definition. However, this coincidence is no longer true for a partial function, since the domain of definition of a partial function can be a proper subset of the domain.

A domain is part of a function f if f is defined as a triple (X, Y, G), where X is called the domain of f, Y its codomain, and G its graph.

A domain is not part of a function f if f is defined as just a graph. For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form f: X → Y.

For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0 (ignoring complex numbers in both cases).

If the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the x-axis. The domain of a function f can be thought of as the set of all x values that can be plugged into f(x) that return a valid output. For example, if we have a function g(x) in the Cartesian plane, the domain is all of the x values such that g(x) is a real number.

Examples:

  • Let be a set of ordered pairs such that . The domain is the set of all x values of , so the domain is .
  • The domain of is all real numbers EXCEPT 0, since 1/0 is not defined.
  • The domain of is , since is only defined when is nonnegative (that is, when is greater than or equal to 0).

Resources and Examples