What is the difference between range and codomain?

Answer 1

Sometimes it is useful to distinguish between the set of images of the domain of a function, which is called the RANGE of the function, from the super-set of the range. For example, for the real function x^2 without any additional qualification, its domain is the set of real numbers and its range is the non-negative reals. The codomain is the entire set of reals that the non-negative reals are part of. So the codomain is the super-set of the range. Curiously, there is no corresponding term for the domain, which might be called the "co-range" but we have no single word for it. If we did, it would simply mean the entire set of which the domain is a subset. So, for example, the domain of the real (otherwise unrestricted) square root of x is the set of non-negative reals and its co-range, if we spoke of it, would be the entire set of reals. We can think of these as the difference between a family and a subset of individuals.

Answer 2

Range of a function f, is the set of attainable points IN the co-domain that f maps elements to.

Co-domain is the space in which the mapped-to elements live.

For example:

say f:R -> R (f maps real numbers to real numbers)

such that

f (t) = e^t

It is true that the co-domain is all real numbers, yet the range is the set of all POSITIVE real numbers, not all the real numbers.