Let $\phi(x,y)$ be an acceptable programming system (i.e., $\phi(x,y)$ is a partial recursive function such that, for every partial recursive function $f(x,y)$, there exists a recursive function $r(x)$ such that, for all $x$ and $y$, $\phi(r(x), y) = f(x,y)$). Is there an acceptable programming system $\psi$ such that, for all $x$, $\mathit{Range}(\phi\_x)=\mathit{Domain}(\psi\_x)$?. (Note that, for all functions $\alpha(x,y)$ and all $x$, $\alpha\_x$ denotes the function $\lambda y.\alpha(x,y)$.)



Range vs Domain of computable functions
The answer is positive.


For each $n$ we can find effectively an index $s(n)$ such that $Range(\phi\_{s(n)})=Domain(\phi\_n)$ and $s(n)>s(m)$ for all $m<n$. Then $Range(s)$ is computable, and we can define
$$
\psi\_x(y)=\begin{cases}\phi\_{n}(y), & \mbox{if }x=s(n);\cr
0,& \mbox{if }x\notin Range(s)\ \& \ y\in Range(\phi\_x);\cr
\uparrow&\mbox{otherwise}.
\end{cases}
$$
Now $\phi\_n=\psi\_{s(n)}$ for all $n$, and $Range(\phi\_x)=Domain(\psi\_x)$ for all $x$.


