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In mathematics, the domain of a function is the set of all input values to the function.
Given a function , the set A 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, or f(A).
A well-defined function must map every element of the domain to an element of its codomain. So, for example, the function:
has no valid value for f(0). It is thus not a function on the set R of real numbers; R can't be its domain. It is usually either defined as a function on R \ {0}, or the "gap" is plugged by specifically defining f(0); for example:
The domain of given function can be restricted to a subset. Suppose that , and . Then the restriction of g to S is written:
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