What is the purpose of having a one-way function also be a permutation on it's on domain? From a cryptographic sense, what value is added from setting the domain to be the image in the mapping?
A one-way permutation (OWP) is more structured than a one-way function, and hence a stronger object. This makes the task of constructing higher cryptographic objects from OWPs easier and also, at times, leads to more efficient constructions. For example, consider pseudo-random generators (PRGs): it is quite straightforward to construct a PRG from a OWP; but constructing a PRG from a OWF took a lot more effort -- you can read more here.