real analysis – Distorted elementary functions


Let $f(x)$ be an elementary function defined on $Xsubseteqmathbb{R}$ and $xi(x), eta(y)$ strictly monotone for $xin X,, yin f(x)$.

Questions:

  • is there an established name for functions of the form $eta(f(xi(x)))$, e.g. if $f(x)$ is a polynomial?
  • are there classes of functions, say polynomials, that can be recognized from a distorted definition, resp., when is it possible to recover $f(x)$ from $eta(f(xi(x)))$ if the type (e.g. polynomial) of $f(x)$ is known?