# if I get the asymptotic solution of a certain equation involving $f(x)$ does it mean that the solution exists

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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