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A function is a mapping from one set - the domain - to another set - the codomain or range - such that each element in the domain is associated with one and only one element in the range.
The domain and codomain need not be different.
It is possible for several elements in the domain to be mapped onto the same element in the range ie a "many-to-one" mapping. However a "one-to-many" mapping not permitted. It may be possible to redefine the domain or range of a one-to-many mapping so that it is no longer is one-to-many and so becomes a function.
For example,
f(x) = x2 (for real x) is a perfectly legitimate many-to-one function. Both -2 and +2 are mapped to 4, but that is OK.
f(x) = sqrt(x) for x ≥ 0 is not a function because 4 can be mapped to -2 or +2. To avoid this, you can restrict the range to f(x) ≥ 0 or define f(x) = |sqrt(x)|.
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