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For (1)(a), a primitive recursive function is obtained by a finite construction from the zero function, successor function and projection functions, using function composition in recursion theory and primitive recursion. The recursion rule, with parameters $\mathbf{x}$, isHere $g,h$ have already been constructed. All these functions are total on nonnegative integers: the initial functions are total, composition preserves totality, and ordinary mathematical induction proves totality of each recursion.

For (1)(b), use a shifted hyperoperation sequence, beginning with addition. Defineand, for each fixed $r\ge 0$, defineThe first three members are therefore addition, multiplication and exponentiation:The exponentiation convention includes $a^0=1$, hence $0^0=1$. The next member iterates exponentiation, with $H_3(a,0)=1$; there is no need for an exceptional domain restriction at $a=0$.

For (1)(c), $H_0$ is a primitive recursive function by its displayed recursion. If $H_r$ is a primitive recursive function, thenare primitive recursive functions, using constants, projection functions and function composition in recursion theory. Primitive recursion applied to $g_r,h_r$ gives $H_{r+1}$. Mathematical induction on the fixed rank $r$ proves every member of the sequence is primitive recursive. This does not assert that the joint three-variable evaluator $(r,a,b)\mapsto H_r(a,b)$ is a primitive recursive function.

For (2), as written, the answer is no: the empty set $X=\varnothing$ is semidecidable, whereas a total computable function from $\mathbb{N}$ to $\mathbb{N}$ always has a nonempty range. This is the only obstruction. To prove the stronger useful statement, suppose $X$ is a nonempty computably enumerable set and choose $x_0\in X$. Fix a Turing machine which halts exactly on the inputs in $X$. Its bounded halting predicateis primitive recursive: encode configurations arithmetically, iterate its total single-step operation $t$ times by primitive recursion, and check the halt state. A halted configuration is kept fixed. This is a bounded simulation, not unbounded waiting.

Fix a primitive recursive pairing function $⟨x,t⟩$ with primitive recursive inverse coordinate functions. For example, the Cantor pairing function has inverses obtained by bounded minimization. DefineThe two coordinate functions, the bounded predicate, and the finite case distinction are primitive recursive functions, so $f$ is a primitive recursive function. Its values always lie in $X$. Conversely, if $x\in X$, its computation halts within some finite $t$, and $f(⟨x,t⟩)=x$. ThusThe range need not be a computable set; deciding whether a value occurs anywhere in this total enumeration is an unbounded search.

A pairing function whose forward map and two inverse coordinate functions are primitive recursive functions. For example, the Cantor pairing function $⟨x,y⟩=(x+y)(x+y+1)/2+y$ has such inverses: find the diagonal index by bounded minimization over integers at most the encoded value, then recover $x,y$ by subtraction. It enables finite tuples to be decoded within a primitive recursive construction.