Effective operations on indices of r.e. sets

Show that for every partial recursive function $f$, there exists a recursive function $g$ such that $W_{g(x)}=f^{-1}\left(W_{x}\right)$.

Show that there is a recursive function $g$ such that for all $x, y$,

Show that there exists a partial recursive function $p: \left\{ 0,1\right\}^{*} \rightarrow \left\{ 0,1\right\}^{*}$ such that $D_{p}=D_{f} \cup D_{g} \cup D_{h}$, and for each $x \in D_{p}$, $p(x)=f(x)$ or $p(x)=g(x)$ or $p(x)=h(x)$.

Show that there does not always exist a partial recursive function $q: \left\{ 0,1\right\}^{*} \rightarrow \left\{ 0,1\right\}^{*}$ such that $D_{q}=D_{f} \cup D_{g} \cup D_{h}$, and for each $x \in D_{q}, q(x)=f(x)$ or $q(x)=g(x)$ or $q(x)=h(x)$, and for each $x \in D_{f} \cap D_{g} \cap D_{h}, q(x)=\min \left\{ f(x), g(x), h(x)\right\}$.

functions defined at the point $0$;

functions $f$ such that $f(a) = b$ for given numbers $a$ and $b$;

functions with nonempty domain of definition.

those containing a given number $a$;

the nonempty ones.

$\pi_{x} \cap \pi_{y} = \pi_{f(x,y)}$;

$\pi_{x} \cup \pi_{y} = \pi_{g(x,y)}$;

$\left\{ x\right\} = \pi_{h(x)}$;

$\left\{ c(s,t) \mid s \in \pi_{x}, t \in \pi_{y}\right\} = \pi_{u(x,y)}$;

$\kappa_{y}(\pi_{x}) = \pi_{v(x,y)}$;

$\kappa_{y}^{-1}(\pi_{x}) = \pi_{w(x,y)}$.

$\delta_{\kappa_{x}} = \rho_{\kappa_{f(x)}}$;

$\rho_{\kappa_{x}} = \delta_{\kappa_{g(x)}}$.