When we can express the inverse of sum of two functions for example f =...

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.