Currently, I am facing this problem:

Given two real functions $A( \vec x )$ and $B( \vec x )$, $R^N\to R$, I want to find a third real, monotonic function f(x), $R\to R$ such that:

$$A(f( \vec x ))=f(B( \vec x ))$$

where simplified the notation writing $f( \vec x )$ meaning: $f(x_1,x_2, ...,x_n) = (f(x_1), f(x_2),..., f(x_n)) $.

I am interested in either having a formula/method for finding f, or even just having a proof that f exists (or doesn't) under some specific conditions. Eventually, I am interested also in the solution in the case N=1.

Also, does this type of problem have a specific name?

Thank you very much!