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Let's take a complicated functional equation $ f(g(x))=f(1-x)g(x) $.

Let us suppose that by using

a) Analytic method

b) Numerical method

I can prove that for example $ f(x) \sim x $ as $ x\rightarrow \infty $; does it mean that I have proved that a solution for the functional equation exists ?

Let's take another problem: I know the function implicitly $f^{-1}(x)= x+d(x)$; here $ d(x)$ is an oscillating function smaller than $x$ for example $d(x) = O(\log x) $ then the approximate solution is $f(x)=x$ in case we ignore the inverse of $d(x) $

Have I proven that the function $x+d(x)$ is invertible? thanks.

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