This a problem I recently dealt with. It's fairly general.
Is the inversion of an invertible map $f\colon X \to Y$, given representatively by $[a] \mapsto [\alpha(a)]$, where $a$ rsp. $\alpha(a)$ represent elements in $X$ rsp. $Y$, – is its inversion $f^{-1}$ given by $[b] \mapsto [\alpha^{-1}(b)]$?, e.g. is $[a] \mapsto [a+1]$ inverted by $[b] \mapsto [b-1]$?
I think I have an answer, but I feel unsure about it. So I answered this question myself which always feels weird, although it's encouraged. Please have a look at it and check that I didn't make any mistakes. I'm also unsure about whether there are some special cases where a formula can't be interpreted as a map of suitable sets with suitable projections.