Domain of a function

A functionffromXtoY.The set of points in the red ovalXis the domain off.

Graph of the real-valuedsquare rootfunction,f(x) =√x,whose domain consists of all nonnegative real numbers

Inmathematics,thedomain of a functionis thesetof inputs accepted by thefunction.It is sometimes denoted by$\operatorname {dom} (f)$or$\operatorname {dom} f$,wherefis the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".

More precisely, given a function$f\colon X\to Y$,the domain offisX.In modern mathematical language, the domain is part of the definition of a function rather than a property of it.

In the special case thatXandYare both sets ofreal numbers,the functionfcan be graphed in theCartesian coordinate system.In this case, the domain is represented on thex-axis of the graph, as the projection of the graph of the function onto thex-axis.

For a function$f\colon X\to Y$,the setYis called thecodomain:the set to which all outputs must belong. The set of specific outputs the function assigns to elements ofXis called itsrangeorimage.The image of f is a subset ofY,shown as the yellow oval in the accompanying diagram.

Any function can be restricted to a subset of its domain. Therestrictionof$f\colon X\to Y$to$A$,where$A\subseteq X$,is written as$\left.f\right|_{A}\colon A\to Y$.

Natural domain

If areal functionfis given by a formula, it may be not defined for some values of the variable. In this case, it is apartial function,and the set of real numbers on which the formula can be evaluated to a real number is called thenatural domainordomain of definitionoff.In many contexts, a partial function is called simply afunction,and its natural domain is called simply itsdomain.

Examples

• The function$f$defined by$f(x)={\frac {1}{x}}$cannot be evaluated at 0. Therefore, the natural domain of$f$is the set of real numbers excluding 0, which can be denoted by$\mathbb {R} \setminus \{0\}$or$\{x\in \mathbb {R}:x\neq 0\}$.
• Thepiecewisefunction$f$defined by$f(x)={\begin{cases}1/x&x\not =0\\0&x=0\end{cases}},$has as its natural domain the set$\mathbb {R}$of real numbers.
• Thesquare rootfunction$f(x)={\sqrt {x}}$has as its natural domain the set of non-negative real numbers, which can be denoted by$\mathbb {R} _{\geq 0}$,the interval$[0,\infty )$,or$\{x\in \mathbb {R}:x\geq 0\}$.
• Thetangent function,denoted$\tan$,has as its natural domain the set of all real numbers which are not of the form${\tfrac {\pi }{2}}+k\pi$for someinteger$k$,which can be written as$\mathbb {R} \setminus \{{\tfrac {\pi }{2}}+k\pi:k\in \mathbb {Z} \}$.

Other uses

The termdomainis also commonly used in a different sense inmathematical analysis:adomainis anon-emptyconnectedopen setin atopological space.In particular, inrealandcomplex analysis,adomainis a non-empty connected open subset of thereal coordinate space$\mathbb {R} ^{n}$or thecomplex coordinate space$\mathbb {C} ^{n}.$

Sometimes such a domain is used as the domain of a function, although functions may be defined on more general sets. The two concepts are sometimes conflated as in, for example, the study ofpartial differential equations:in that case, adomainis the open connected subset of$\mathbb {R} ^{n}$where a problem is posed, making it both an analysis-style domain and also the domain of the unknown function(s) sought.

Set theoretical notions

For example, it is sometimes convenient inset theoryto permit the domain of a function to be aproper classX,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 formf:X→Y.

See also

• Argument of a function
• Attribute domain
• Bijection, injection and surjection
• Codomain
• Domain decomposition
• Effective domain
• Image (mathematics)
• Lipschitz domain
• Naive set theory
• Range of a function
• Support (mathematics)

References

• Bourbaki, Nicolas (1970).Théorie des ensembles.Éléments de mathématique. Springer.ISBN9783540340348.
• Eccles, Peter J. (11 December 1997).An Introduction to Mathematical Reasoning: Numbers, Sets and Functions.Cambridge University Press.ISBN978-0-521-59718-0.
• Mac Lane, Saunders (25 September 1998).Categories for the Working Mathematician.Springer Science & Business Media.ISBN978-0-387-98403-2.
• Scott, Dana S.; Jech, Thomas J. (31 December 1971).Axiomatic Set Theory, Part 1.American Mathematical Soc.ISBN978-0-8218-0245-8.
• Sharma, A. K. (2010).Introduction To Set Theory.Discovery Publishing House.ISBN978-81-7141-877-0.
• Stewart, Ian; Tall, David (1977).The Foundations of Mathematics.Oxford University Press.ISBN978-0-19-853165-4.