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Domain (function))
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In mathematics, the domain of a function is the set of all input values to the function.
Given a function <math>f\colon A\rightarrow B<math>, 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:
- <math>f\colon\,x\mapsto 1/x<math>
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:
- <math>f\colon x\mapsto 1/x,\quad x\neq0<math>
- <math>f\colon0\mapsto 0<math>
The domain of given function can be restricted to a subset.
Suppose that <math>g\colon A\to B<math>, and <math>S\subset A<math>.
Then the restriction of g to S is written:
- <math>g|_S\colon S\to B<math>
See also
codomain,
range (mathematics),
injective function,
surjective function,
bijective function
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