Illustration showing
f, a function from pink domain
X to blue and yellow codomain
Y. The smaller yellow oval inside
Y is the
image of
f. Either the image or the codomain also sometimes is called the
range of
f.
In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain.^{[1]} Conversely, the set of values the function takes is termed the image of the function, which is sometimes also referred to as the range of the function.
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). When the domain of a function is a subset of the real numbers, and the function is represented in an xy Cartesian coordinate system, the domain is represented on the xaxis.
Formal definition
Given a function f:X→Y, the set X is the domain of f; the set Y is the codomain of f. In the expression f(x), x is the argument and f(x) is the value. One can think of an argument as an input to the function, and the value as the output.
The image (sometimes called the range) of f is the set of all values assumed by f for all possible x; this is the set {f(x)  x ∈ X}. The image 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; it is the whole codomain if and only if f is a surjective function.
A welldefined 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 all real numbers, R, cannot be its domain. In cases like this, the function is either defined on R−{0} or the "gap is plugged" by explicitly defining f(0). If we extend the definition of f to

f(x) = \begin{cases} 1/x&x\not=0\\ 0&x=0 \end{cases}
then f is defined for all real numbers, and its domain is \mathbb{R}.
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.
Natural domain
The natural domain of a function is the set of values for which it is defined, typically within the reals but sometimes among the integers or complex numbers. For instance the natural domain of square root is the nonnegative 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.^{[2]}
Domain of a partial function
There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function from X to Y, i.e. a function from a subset X' of X to Y. Most mathematicians, including recursion theorists, use the term "domain of f" for the set X' of all values x such that f(x) is defined. But some, particularly category theorists, consider the domain to be X, irrespective of whether f(x) exists for every x in X.
Category theory
In category theory 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 for more.
Real and complex analysis
In real and complex analysis, a domain is an open connected subset of a real or complex vector space.
In partial differential equations, a domain is an open connected subset of the euclidean space R^{n}, where the problem is posed, i.e., where the unknown function(s) are defined.
More examples

As a partial function from the real numbers the function x\mapsto \sqrt{x} is defined for all x \geq 0.

If one defines the square root of a negative number x as the complex number z, with positive imaginary part, such that z^{2} = x, the function x\mapsto \sqrt{x} is defined for all real numbers x.

Function \tan x = \frac{\sin x}{\cos x} is defined for all x \neq \frac{\pi}{2} + k \pi, k = 0, \pm 1, \pm 2, \ldots
See also
References

^ Paley, H. Abstract Algebra, Holt, Rinehart and Winston, 1966 (p. 16).

^ Rosenbaum, Robert A.; Johnson, G. Philip (1984). Calculus: basic concepts and applications. Cambridge University Pressd. p. 60.
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