Your g is not reversible though, its input space must be the same dimension than its output space. On the other hand, you're correct in that any irreversible function can be extended to a reversible one, although the process isn't always straightforward. The general way is to do somthing like:

f: V -> V

g: V x A -> V x A

with g(x, a) = g(f(x), b) for some value(s) of a. And b cannot be set to x, because then you can't find a back with g', and your function is not invertible.