I will rewrite my qusting using Matt F. suggestion.

Consider $R$ in the language $L$ with one function $f$, and a family of relations like $$\{Sq(x,y):=x=y^2, In(x):=x∈[1,3]\}$$ Consider the map $Q:2^\mathbb{R}→2^\mathbb{R}$ by $$Q(S)=\{a:∃x,y∈S \: In(a)∧a=f(x)∧Sq(x,y)\}$$ which has a first-order definition in $L$ and uses $f$ only once. For a given language with one function $f$ and a family of relations, can we characterize the maps from $2^\mathbb{R}→2^\mathbb{R}$ which have similar first-order definitions in $L$ and use $f$ only once?

Motivation: I wanted to give logically correct definition of all constructions of fractals such as Apollonian gasket and Sierpinski triangle. In the setting of Apollonian gasket it is natural to replace $\mathbb{R}$ with the set $\mathcal{S}$ of circles in $\mathbb{R}^2$ and $f$ with the map that inputs three pairwase tangent circles and outputs two tangent circles to a given three ones. So if we want to make a map $F$ that correctly add tangent circles to $i$-th step of Apollonian gasket we need some linear logical expressed function from $f$.