# Is there any branch of constructing inverse function exist that

Is there any branch of constructing inverse function exist that is inverse of $n$ functions?
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Let be functions with domain for a set $I$, s.t. the sets ${D}_{i}$ are pairwise disjoint. Then you can construct a function $f$ s.t. . If $f$ is bijective, an inverse function ${f}^{-1}$ exists.
It then holds that ${f}^{-1}{|}_{{f}_{i}\left({D}_{i}\right)}={f}_{i}^{-1}$.