Given computable function, what are conditions for computability of inverse function?
Let us say that a computable function $f$ is invertible if there is another computable function $g$ that on input $y$ either finds $x$ such that $f(x) = y$ or returns $\bot$ when $y$ has no preimage.

For this definition, one can show that a computable function $f$ is invertible if and only if its range is decidable, that is, we can decide whether a given input has a preimage under $f$.