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

Question

The function $\mathrm{g}$, with domain the non-negative integers, is defined by $$\mathrm{g}(n)=\begin{cases}1,&n=0,\2\mathrm{g}\left(\dfrac n2\right),&n\text{ is positive and even},\1+\mathrm{g}\left(\dfrac{n-1}{2}\right),&n\text{ is odd}.\end{cases}$$ Find $\mathrm{g}(7)$, $\mathrm{g}(10)$ and $\mathrm{g}(12)$.[3] Find all values of $n$ for which $\mathrm{g}(n)=4$, showing your reasoning.[3] Hence explain why $\mathrm{g}$ does not have an inverse.[2]

Accepted Answer

(a) $4,10,12$ (b) $n=2$ or $n=7$. (c) $\mathrm{g}(2)=\mathrm{g}(7)=4$, so $\mathrm{g}$ is not one-to-one.

Working backwards through a recursive rule

Working backwards through a recursive rule

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