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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the domain (or replacement set) of a given function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 is the set of "input
Input

Input is the term denote either an entrance or changes which are inserted into a system and which activate/modify a process. It is an abstract concept, used in the model ing, system design and system exploitation....
" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root
Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
 would be only numbers greater than or equal to 0 (ignoring complex numbers in both cases). In a representation of a function in a xy Cartesian coordinate system
Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
, the domain is represented on the x axis (or abscissa).

n a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 f:X?Y, the set X of input values is the domain of f; the set Y is the codomain
Codomain

In mathematics, the codomain, range or target set, of a function , described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall....
 of f.

The range
Range (mathematics)

In mathematics, the range of a function is the Set of all "output" values produced by that function. Sometimes it is called the , or more precisely, the image of the domain of the function....
 of f is the set of all output values of f; this is the set .






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the domain (or replacement set) of a given function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 is the set of "input
Input

Input is the term denote either an entrance or changes which are inserted into a system and which activate/modify a process. It is an abstract concept, used in the model ing, system design and system exploitation....
" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root
Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
 would be only numbers greater than or equal to 0 (ignoring complex numbers in both cases). In a representation of a function in a xy Cartesian coordinate system
Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
, the domain is represented on the x axis (or abscissa).

Formal definition

Given a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 f:X?Y, the set X of input values is the domain of f; the set Y is the codomain
Codomain

In mathematics, the codomain, range or target set, of a function , described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall....
 of f.

The range
Range (mathematics)

In mathematics, the range of a function is the Set of all "output" values produced by that function. Sometimes it is called the , or more precisely, the image of the domain of the function....
 of f is the set of all output values of f; this is the set . The range of f can be the same set as the codomain or it can be a proper subset of it. It is in general smaller than the codomain unless f is a surjective function
Surjective function

In mathematics, a function f is said to be surjective or onto, if its values span its whole codomain; that is, for every y in the codomain, there is at least one x in the domain such that f = y ....
.

A well defined function must map every element of its domain to an element of its codomain. For example, the function f defined by
f(x) = 1/x
has no value for f(0). Thus, the set of real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s, , cannot be its domain. In cases like this, the function is either defined on or the "gap is plugged" by explicitly defining f(0). If we extend the definition of f to
f(x) = 1/x, for x ? 0
f(0) = 0,
then f is defined for all real numbers, and its domain is .

Any function can be restricted to a subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 of its domain. The restriction of g : A ? B to S, where S ? A, is written g |S : S ? B.

Domain of a partial function

There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function
Partial function

In mathematics, a partial function is a binary relation that associates each element of a Set , sometimes called its domain , with at most one element of another set, called its codomain....
. Most mathematicians, including recursion theorists
Recursion theory

Recursion theory, also called computability theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees....
, use the term "domain of f" for the set of all values x such that f(x) is defined. But some, particularly category theorists
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
, consider the domain of a partial function f:X?Y to be X, irrespective of whether f(x) exists for every x in X.

Category theory


In category theory
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
 one deals with morphisms instead of functions. Morphisms are arrows from one object to another. The domain of any morphism is the object from which an arrow starts. In this context, many set theoretic ideas about domains must be abandoned or at least formulated more abstractly. For example, the notion of restricting a morphism to a subset of its domain must be modified. See subobject
Subobject

In category theory, a branch of mathematics, a subobject is, roughly speaking, an object which sits inside another object in the same category ....
 for more.

Real and complex analysis


In real
Calculus

Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
 and complex analysis
Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating Function of complex numbers....
, a domain is an open
Open set

In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
 connected
Connected space

In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open subsets....
 subset of a real
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
 or complex
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
 vector space.

In partial differential equation
Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a Relation involving an unknown Function of several independent variables and its partial derivatives with respect to those variables....
s, a domain is an open connected subset of the euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
 Rn
RN

RN is a journal for registered nurses.RN may stand for:* Registered Nurse* Bullet#Bullet Abbreviations bullet* Royal Navy* Radio National, an Australia-wide radio network broadcast by the Australian Broadcasting Corporation...
, where the problem is posed, i.e., where the unknown function(s) are defined.

See also


  • Range (mathematics)
    Range (mathematics)

    In mathematics, the range of a function is the Set of all "output" values produced by that function. Sometimes it is called the , or more precisely, the image of the domain of the function....
  • Codomain
    Codomain

    In mathematics, the codomain, range or target set, of a function , described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall....
  • Surjective function
    Surjective function

    In mathematics, a function f is said to be surjective or onto, if its values span its whole codomain; that is, for every y in the codomain, there is at least one x in the domain such that f = y ....
  • Injective function
    Injective function

    In mathematics, an injective function is a function which associates distinct arguments with distinct values.An injective function is called an injection, and is also said to be a one-to-one function ....
  • Bijection
    Bijection

    In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y....
  • Domain decomposition
  • Lipschitz domain
    Lipschitz domain

    In mathematics, a Lipschitz domain is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuity....