Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 12G / c / Solution

Let $S(n)=n+1$ be the successor function. Define $f$ by primitive recursionInduction gives $f(n)=2n$. Then defineby composition. Initial functions, composition, and primitive recursion produce primitive recursive functions, so $f$ and $g$ are partial recursive functions and are visibly defined for every $n$. Both functions are total.