In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by \operatorname{dom}(f) or \operatorname{dom }f, where f is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".[1]

More precisely, given a function f\colon X\to Y, the domain of f is X. In modern mathematical language, the domain is part of the definition of a function rather than a property of it.

In the special case that X and Y are both sets of real numbers, the function f can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the x-axis of the graph, as the projection of the graph of the function onto the x-axis.

For a function f\colon X\to Y, the set Y is called the codomain: the set to which all outputs must belong. The set of specific outputs the function assigns to elements of X is called its range or image. The image of f is a subset of Y, shown as the yellow oval in the accompanying diagram.

Any function can be restricted to a subset of its domain. The restriction of f \colon X \to Y to A, where A\subseteq X, is written as \left. f \right|_A \colon A \to Y.

Natural domain

If a real function f is given by a formula, it may be not defined for some values of the variable. In this case, it is a partial function, and the set of real numbers on which the formula can be evaluated to a real number is called the natural domain or domain of definition of f. In many contexts, a partial function is called simply a function, and its natural domain is called simply its domain.

Examples

  • The function f defined by f(x)=\frac{1}{x} cannot be evaluated at 0. Therefore, the natural domain of f is the set of real numbers excluding 0, which can be denoted by \mathbb{R} \setminus \{ 0 \} or \{x\in\mathbb R:x\ne 0\}.
  • The piecewise function f defined by f(x) = \begin{cases} 1/x&x\not=0\\ 0&x=0 \end{cases}, has as its natural domain the set \mathbb{R} of real numbers.
  • The square root function f(x)=\sqrt x has as its natural domain the set of non-negative real numbers, which can be denoted by \mathbb R_{\geq 0}, the interval [0,\infty), or \{x\in\mathbb R:x\geq 0\}.
  • The tangent function, denoted \tan, has as its natural domain the set of all real numbers which are not of the form \tfrac{\pi}{2} + k \pi for some integer k, which can be written as \mathbb R \setminus \{\tfrac{\pi}{2}+k\pi: k\in\mathbb Z\}.

Other uses

The term domain is also commonly used in a different sense in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis, a domain is a non-empty connected open subset of the real coordinate space \R^n or the complex coordinate space \C^n.

Sometimes such a domain is used as the domain of a function, although functions may be defined on more general sets. The two concepts are sometimes conflated as in, for example, the study of partial differential equations: in that case, a domain is the open connected subset of \R^{n} where a problem is posed, making it both an analysis-style domain and also the domain of the unknown function(s) sought.

Set theoretical notions

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.[2]

See also

Notes

  1. ^ "Domain, Range, Inverse of Functions". Easy Sevens Education. 10 April 2023. Retrieved 2023-04-13.
  2. ^ Eccles 1997, p. 91 ([ quote 1], [ quote 2]); Mac Lane 1998, [ p. 8]; Mac Lane, in Scott & Jech 1971, [ p. 232]; Sharma 2010, [ p. 91]; Stewart & Tall 1977, [ p. 89]

References