Domain (mathematics)

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. Conversely, the set of values the function takes on as output 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).

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.

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 a member of the domain that is chosen as an "input" to the function, and the value as the "output" when the function is applied to that member of the domain.

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.

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