The function $\mathrm{f}$ is given by $\mathrm{f}:x\mapsto 1+\sqrt{x}$, for \(x\in \mathbb{

Question

The function $\mathrm{f}$ is given by $\mathrm{f}:x\mapsto 1+\sqrt{x}$, for $x\in \mathbb{R}$, $x\ge 0$. Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and state the domain of ${{\mathrm{f}}^{-1}}$.[3] Show that if $\mathrm{ff}\left( x \right)=x$ then ${{x}^{3}}-4{{x}^{2}}+4x-1=0$. Hence find the value of $x$ for which $\mathrm{ff}\left( x \right)=x$. Explain why this value of $x$ satisfies the equation $\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$.[5] The function $\mathrm{g}$, with domain the set of non-negative integers, is given by $g\left( n \right)=\left\{ \begin{matrix} 1\mathrm{for}\,n=0, \ 2+\mathrm{g}\left( \frac{1}{2}n \right)\mathrm{for \; }n\mathrm{ \; even,} \ 1+\mathrm{g}\left( n-1 \right)\mathrm{for \; }n\mathrm{ \; odd}. \ \end{matrix} \right.$ Find $\mathrm{g}\left( 4 \right)$, $\mathrm{g}\left( 7 \right)$and $\mathrm{g}\left( 12 \right)$.[3] Does $\mathrm{g}$ have an inverse? Justify your answer.[2]

Accepted Answer

(b)(i) $\mathrm{g}\left( 4 \right)=6$, $\mathrm{g}\left( 7 \right)=8$ and $\mathrm{g}\left( 12 \right)=9$. (ii) $\mathrm{g}$ is not $1-1$. Therefore, it doesn’t have an inverse.

N2016 P1 Q10

N2016 P1 Q10

Video Solution:

Default solution