An arbitrary function's image can't in general be determined, so a "range" (by which I mean here codomain) must be specified -- just as the domain must be -- as part of the definition of the concept of a function. It is unfortunate that authors of calculus books got "range" and "image" so confused that new terminology (codomain) had to be invented. (Though maybe it's good terminology from the point of view of category theory.) It took a few centuries to hammer out the correct definition of a function: A subset f of the cartesian product XxY of the domain X and codomain Y such that for each x in X there is exactly one y in Y such that (x,y) belongs to f. This should not be forgotten or tampered with. --Dan "Things are seldom what they seem." --W.S. Gilbert