The function f(x) = √x has a domain of all numbers between 0 and positive infinity

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Input is the term denoting either an entrance or changes which are inserted into a System and which activate/modify a Process. ^[1] For instance, the domain of cosine would be all real numbers, while the domain of the square root would only be numbers greater than or equal to 0 (ignoring complex numbers in both cases). In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In a representation of a function in a xy Cartesian coordinate system, the domain is represented on the x axis (or abscissa). In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

Formal definition

Given a function f:X→Y, the set X of input values is the domain of f; the set Y is the codomain of f. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the codomain, or target, of a function f: X → Y is the set

The range of f is the set of all output values of f; this is the set . In Mathematics, the range of a function is the set of all "output" values produced by that function ^[2] 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. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every

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 numbers, , cannot be its domain. In Mathematics, the real numbers may be described informally in several different ways 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 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. Domain of a partial function There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function Most mathematicians, including recursion theorists, use the term "domain of f" for the set of all values x such that f(x) is defined. Recursion theory, also called computability theory, is a branch of Mathematical logic that originated in the 1930s with the study of Computable functions But some, particularly category theorists, consider the domain of a partial function f:X→Y to be X, irrespective of whether f(x) exists for every x in X. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

Category theory

In category theory one deals with morphisms instead of functions. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and 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 for more. In Category theory, there is a general definition of subobject extending the idea of Subset and Subgroup.

Real and complex analysis

In real and complex analysis, a domain is an open connected subset of a real or complex vector space. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

See also

• Range (mathematics)
• Codomain
• Surjective function
• Injective function
• Bijection

References

1. ^ Paley, H. In Mathematics, the range of a function is the set of all "output" values produced by that function In Mathematics, the codomain, or target, of a function f: X → Y is the set In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Abstract Algebra, Holt, Rinehart and Winston, 1966 (p. 16).
2. ^ Smith, William K. Inverse Functions, MacMillan, 1966 (p. 8).