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f : A →B is a function if for any element of set A there exists one and only one element of set B . Knowledge of domain and range and the strict compliance of the definition would separate functions from mere mathematical expressions.
A couple of examples: f: R→R with f(x)=x2 i is a function; x2 + y2 = 1 is not a function. It boils down to the strict definition of a function!