Domain of a function
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. More precisely, given a function f\colon X\to Y, the domain of is . Note that 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 and are both subsets of \R, the function can be graphed in the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
. In this case, the domain is represented on the -axis of the graph, as the projection of the graph of the function onto the -axis. For a function f\colon X\to Y, the set is called the codomain, and the set of values attained by the function (which is a subset of ) is called its range or
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
. Any function can be restricted to a subset of its domain. The
restriction Restriction, restrict or restrictor may refer to: Science and technology * restrict, a keyword in the C programming language used in pointer declarations * Restriction enzyme, a type of enzyme that cleaves genetic material Mathematics and log ...
of f \colon X \to Y to A, where A\subseteq X, is written as \left. f \_A \colon A \to Y.


Natural domain

If a real function 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 . 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 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 \setminus \ or \. * The piecewise function f defined by f(x) = \begin 1/x&x\not=0\\ 0&x=0 \end, has as its natural domain the set \mathbb 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_, the interval ,\infty), or \. * The tangent function, denoted \tan, has as its natural domain the set of all real numbers which are not of the form \tfrac + k \pi for some integer k, which can be written as \mathbb R \setminus \.


Other uses

The word "domain" is used with other related meanings in some areas of mathematics. In topology, a domain is a connected open set. In real and
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, a domain is an open connected subset of a real or complex vector space. In the study of
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s, a domain is the open connected subset of the Euclidean space \mathbb^ where a problem is posed (i.e., where the unknown function(s) are defined).


Set theoretical notions

For example, it is sometimes convenient in set theory to permit the domain of a function to be a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
, in which case there is formally no such thing as a triple . With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form ., p. 91 ( quote 1 quote 2; , p. 8 Mac Lane, in , p. 232 , p. 91 , p. 89/ref>


See also

* Attribute domain * Bijection, injection and surjection * Codomain *
Domain decomposition In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solu ...
* Effective domain * Image (mathematics) * Lipschitz domain *
Naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike Set theory#Axiomatic set theory, axiomatic set theories, which are defined using Mathematical_logic#Formal_logical_systems, forma ...
* Support (mathematics)


Notes


References

* {{Mathematical logic Functions and mappings Basic concepts in set theory