Saying right now: Yes, this is homework. I'm not asking for an answer, but I would love any help into a general direction to look at this problem at. I've been working on it now for hours and have not made any real progress.
Can a function, with a well defined inverse, be implemented to map 32 bit integers to 64 bit integers. Do all functions from 32bit to 64bit integers have well defined inverses?
Of course not.
Take the identity function for example. All 32-bit values have an identity in the 64-bit value space (just use 0 in the top 32 bits, using only the bottom 32 bits for the value). However, any 64-bit value where the top 32 bits is not 0, will not have a corresponding value in the 32-bit value space.
The above is a layman's explanation, and is probably not rigorous enough as a homework solution (as intended). You'd do well to read up on the pigeonhole principle.