In mathematics, the domain of a function is the set of all input values to the function.
Given a functionf : A → B, the setA is called the domain, or domain of definition of f.
The set of all values in the codomain that f maps to is called the range of f, written f(A).
A well-defined function must map every element of the 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 R of real numbers cannot be its domain. In cases like this, the function is usually either defined on R {0}, or the "gap" is plugged by specifically 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 we can choose its domain to be 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.
In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range).
A very common type of function occurs when the argument (input) and the value (output) are both numbers, the functional relationship is expressed by a formula, and the value (output) of the function is obtained by direct substitution of the argument into the formula.
The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers.
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