The function $\mathrm{f}$, with domain the non-negative integers, is defined by \[\mathrm{f}(n)=\b

Question

The function $\mathrm{f}$, with domain the non-negative integers, is defined by $$\mathrm{f}(n)=\begin{cases}2,&n=0,\3+\mathrm{f}\left(\dfrac n2\right),&n\text{ is positive and even},\1+\mathrm{f}(n-1),&n\text{ is odd}.\end{cases}$$ Find $\mathrm{f}(6)$, $\mathrm{f}(11)$ and $\mathrm{f}(20)$.[3] Does $\mathrm{f}$ have an inverse? Justify your answer.[2] Find an expression for $\mathrm{f}(2^m)$ in terms of $m$, where $m$ is a non-negative integer.[2]

Accepted Answer

(a) $10,14,16$ (b) No; $\mathrm{f}(5)=\mathrm{f}(6)=10$. (c) $\mathrm{f}(2^m)=3m+3$.

Evaluating a recursive rule and its powers-of-two inputs

Evaluating a recursive rule and its powers-of-two inputs

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