Domain of a function

A function Template:Mvar from Template:Mvar to Template:Mvar. The set of points in the red oval Template:Mvar is the domain of Template:Mvar.

File:Square root 0 25.svg

Graph of the real-valued square root function, f(x) = Template:Radic, whose domain consists of all nonnegative real numbers

In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by <math>\operatorname{dom}(f)</math> or <math>\operatorname{dom }f</math>, where Template:Math is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

More precisely, given a function <math>f\colon X\to Y</math>, the domain of Template:Math is Template:Math. 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 Template:Math and Template:Math are both sets of real numbers, the function Template:Math can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the Template:Math-axis of the graph, as the projection of the graph of the function onto the Template:Math-axis.

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

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

Natural domainEdit

If a real function Template:Mvar 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 Template:Mvar. In many contexts, a partial function is called simply a function, and its natural domain is called simply its domain.

ExamplesEdit

The function <math>f</math> defined by <math>f(x)=\frac{1}{x}</math> cannot be evaluated at 0. Therefore, the natural domain of <math>f</math> is the set of real numbers excluding 0, which can be denoted by <math>\mathbb{R} \setminus \{ 0 \}</math> or <math>\{x\in\mathbb R:x\ne 0\}</math>.
The piecewise function <math>f</math> defined by <math>f(x) = \begin{cases}

1/x&x\not=0\\ 0&x=0 \end{cases},</math> has as its natural domain the set <math>\mathbb{R}</math> of real numbers.

The square root function <math>f(x)=\sqrt x</math> has as its natural domain the set of non-negative real numbers, which can be denoted by <math>\mathbb R_{\geq 0}</math>, the interval <math>[0,\infty)</math>, or <math>\{x\in\mathbb R:x\geq 0\}</math>.
The tangent function, denoted <math>\tan</math>, has as its natural domain the set of all real numbers which are not of the form <math>\tfrac{\pi}{2} + k \pi</math> for some integer <math>k</math>, which can be written as <math>\mathbb R \setminus \{\tfrac{\pi}{2}+k\pi: k\in\mathbb Z\}</math>.

Other usesEdit

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 <math>\R^n</math> or the complex coordinate space <math>\C^n.</math>

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 <math>\R^{n}</math> where a problem is posed, making it both an analysis-style domain and also the domain of the unknown function(s) sought.

Set theoretical notionsEdit

For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class Template:Mvar, in which case there is formally no such thing as a triple Template:Math. With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form Template:Math.<ref>Template:Harvnb, p. 91 ([[[:Template:Google books]] quote 1], [[[:Template:Google books]] quote 2]); Template:Harvnb, [[[:Template:Google books]] p. 8]; Mac Lane, in Template:Harvnb, [[[:Template:Google books]] p. 232]; Template:Harvnb, [[[:Template:Google books]] p. 91]; Template:Harvnb, [[[:Template:Google books]] p. 89]</ref>

NotesEdit

Template:Reflist

ReferencesEdit

Template:Mathematical logic