Domain of a function

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In mathematics, the domain or set of departure of a function is the set into which all of the input of the function is constrained to fall. [1] It is the set X in the notation f: XY, and is alternatively denoted as ${\displaystyle \operatorname {dom} (f)}$. [2] Since a (total) function is defined on its entire domain, its domain coincides with its domain of definition. [3] However, this coincidence is no longer true for a partial function, since the domain of definition of a partial function can be a proper subset of the domain.

Contents

A domain is part of a function f if f is defined as a triple (X, Y, G), where X is called the domain of f, Y its codomain , and G its graph . [4]

A domain is not part of a function f if f is defined as just a graph. [5] [6] For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form f: XY. [7]

For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0 (ignoring complex numbers in both cases).

If the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the x-axis.

Examples

A well-defined function must map every element of its domain to an element of its codomain. For example, the function ${\displaystyle f}$ defined by

${\displaystyle f(x)={\frac {1}{x}}}$

has no value for ${\displaystyle f(0)}$. Thus the set of all real numbers, ${\displaystyle \mathbb {R} }$, cannot be its domain. In cases like this, the function is either defined on ${\displaystyle \mathbb {R} \setminus \{0\}}$, or the "gap is plugged" by defining ${\displaystyle f(0)}$ explicitly. For example. if one extends the definition of ${\displaystyle f}$ to the piecewise function

${\displaystyle f(x)={\begin{cases}1/x&x\not =0\\0&x=0\end{cases}}}$

then ${\displaystyle f}$ is defined for all real numbers, and its domain is ${\displaystyle \mathbb {R} }$.

Any function can be restricted to a subset of its domain. The restriction of ${\displaystyle g\colon A\to B}$ to ${\displaystyle S}$, where ${\displaystyle S\subseteq A}$, is written as ${\displaystyle \left.g\right|_{S}\colon S\to B}$.

Natural domain

The natural domain of a function (sometimes shortened as domain) is the maximum set of values for which the function is defined, typically within the reals but sometimes among the integers or complex numbers as well. For instance, the natural domain of square root is the non-negative reals when considered as a real number function. When considering a natural domain, the set of possible values of the function is typically called its range. [8] Also, in complex analysis especially several complex variables, when a function f is holomorpic on the domain ${\displaystyle D\subset \mathbb {C} ^{n}}$ and cannot directly connect to the domain outside D, including the point of the domain boundary ${\displaystyle \partial D}$, in other words, such a domain D is a natural domain in the sense of analytic continuation, the domain D is called the domain of holomorphy of f and the boundary is called the natural boundary of f.

Category theory

Category theory 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. For more, see subobject.

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. [9] In real and complex analysis, a domain is an open connected subset of a real or complex vector space. In the study of partial differential equations, a domain is the open connected subset of the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ where a problem is posed (i.e., where the unknown function(s) are defined).

More common examples

As a partial function from the real numbers to the real numbers, the function ${\displaystyle x\mapsto {\sqrt {x}}}$ has domain ${\displaystyle x\geq 0}$. However, if one defines the square root of a negative number x as the complex number z with positive imaginary part such that z2 = x, then the function ${\displaystyle x\mapsto {\sqrt {x}}}$ has the entire real line as its domain (but now with a larger codomain). The domain of the trigonometric function ${\displaystyle \tan x={\tfrac {\sin x}{\cos x}}}$ is the set of all (real or complex) numbers, that are not of the form ${\displaystyle {\tfrac {\pi }{2}}+k\pi ,k=0,\pm 1,\pm 2,\ldots }$.

Notes

1. Codd, Edgar Frank (June 1970). "A Relational Model of Data for Large Shared Data Banks" (PDF). Communications of the ACM. 13 (6): 377–387. doi:10.1145/362384.362685 . Retrieved 2020-04-29.
2. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-28.
3. Paley, Hiram; Weichsel, Paul M. (1966). . New York: Holt, Rinehart and Winston. p.  16.
4. Bourbaki 1970 , p. 76
5. Bourbaki 1970 , p. 77
6. Forster 2003 , pp. 1011
7. Eccles 1997 , p. 91 (quote 1, quote 2); Mac Lane 1998 , p. 8; Mac Lane, in Scott & Jech 1967 , p. 232; Sharma 2004 , p. 91; Stewart & Tall 1977 , p. 89
8. Rosenbaum, Robert A.; Johnson, G. Philip (1984). . Cambridge University Press. p.  60. ISBN   0-521-25012-9.
9. Weisstein, Eric W. "Domain". mathworld.wolfram.com. Retrieved 2020-08-28.

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References

• Bourbaki, Nicolas (1970). Théorie des ensembles. Éléments de mathématique. Springer. ISBN   9783540340348.