The domain of a function f:AB is A.

This is an important point to make: the domain is not the set of values for which an interpretation of the function may make sense. Take, for example, f(x) = (2x + 3). This, believe it or not, is not a full mathematical definition for a function. A more full definition would be f(x):ℜ → ℜ = (2x + 3). In this case, the domain is ℜ, or the set of real numbers. A different function might be g(x):QQ = (2x + 3) [Q being the set of rational numbers]. The value of g(x) is equal to the value of f(x), when x is in the domain of both functions, but f and g are in fact different functions; for example, f(√2) is defined but g(√2) is undefined [because the square root of 2 is not a rational number and so not in the domain of g]. Other similar functions might have domains of the integers, or complex numbers, or multiples of √5, or integers modulo n, or p-adic integers, or other sets…

It often happens that the domain of a function is not specified, and in those cases, the obvious logical choice is usually good enough for the task at hand. However, it does make a difference in some special cases, and the strict definition of the domain becomes important.

Compare with range and codomain.