Editing Domain of a function (section)

{{short description|Mathematical concept}}
[[File:Codomain2.SVG|right|thumb|250px|A function {{mvar|f}} from {{mvar|X}} to {{mvar|Y}}. The set of points in the red oval {{mvar|X}} is the domain of {{mvar|f}}.]]
[[File:Square_root_0_25.svg|thumb|250px|Graph of the real-valued [[square root]] function, ''f''(''x'') = {{radic|''x''}}, whose domain consists of all nonnegative real numbers]]

In [[mathematics]], the '''domain of a function''' is the [[Set (mathematics)|set]] of inputs accepted by the [[Function (mathematics)|function]]. It is sometimes denoted by <math>\operatorname{dom}(f)</math> or <math>\operatorname{dom }f</math>, where {{math|''f''}} is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".<ref>{{Cite web|title=Domain, Range, Inverse of Functions|url=https://www.easysevens.com/domain-range-inverse-of-functions/|access-date=2023-04-13|website=Easy Sevens Education|date=10 April 2023 |language=en}}</ref>

More precisely, given a function <math>f\colon X\to Y</math>, the domain of {{math|''f''}} is {{math|''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 {{math|''X''}} and {{math|''Y''}} are both sets of [[real number]]s, the function {{math|''f''}} can be graphed in the [[Cartesian coordinate system]]. In this case, the domain is represented on the {{math|''x''}}-axis of the graph, as the projection of the graph of the function onto the {{math|''x''}}-axis.

For a function <math>f\colon X\to Y</math>, the set {{math|''Y''}} is called the ''[[codomain]]'': the set to which all outputs must belong. The set of specific outputs the function assigns to elements of {{math|''X''}} is called its ''[[Range of a function|range]]'' or ''[[Image (mathematics)|image]]''. The image of f is a subset of {{math|''Y''}}, shown as the yellow oval in the accompanying diagram.

Any function can be restricted to a subset of its domain. The [[Restriction (mathematics)|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>.