Jeffery Clements

2022-06-25

When we can express the inverse of sum of two functions for example $f={f}_{1}+{f}_{2}$ in terms of inverse of two functions $\left({f}_{1}^{-1},{f}_{2}^{-1}\right)$?

drumette824ed

Expert

There is no easy way to do so. (There is an obvious hard way: invert ${f}_{1}^{-1}$ and ${f}_{2}^{-1}$ to get ${f}_{1}$ and ${f}_{2}$, sum these to get $f$ and then invert.)
It can well happen that ${f}_{1}$ and ${f}_{2}$ are invertible but $f$ is not. Or that neither of ${f}_{1}$ and ${f}_{2}$ is invertible but $f$ is. And you can find an invertible function ${f}_{1}$ and a non-invertible function ${f}_{2}$ so that $f$ is invertible — and another pair of such functions ${f}_{1}$ and ${f}_{2}$ so that $f$ is not invertible.

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