Iteration of a partial function ($D_{n+1}=\{x\in D_n:f^n(x)\in D\}$)

For a partial function $f:D\to X$ with $D\subseteq X$, its $n$th iterated function is defined only when each intermediate application is defined. Put $D_0=X$, $f^0={\mathrm{id}}_X$, and use the displayed recursion for the domain of a function. Simplifying an algebraic expression for $f^n$ does not restore points excluded by an intermediate application. For example, $f(x)=1/x$ has $f^2(x)=x$ on $\mathbb{R}\setminus \{0\}$; the simplified expression $x$ has a larger natural domain of a function.