2018 NJC P1 Q6(b)

The function \(\mathrm{g}\) is defined by

\(\mathrm{g}\left( x \right)=\left\{ \begin{matrix}<br />& {{\left( x-1 \right)}^{2}},\,\,\,\,\,\,\mathrm{for}\,0\le x\le 1, \\ <br />& 2{{\log }_{2}}x,\,\,\,\,\,\,\,\mathrm{for}\,1<x\le 2. \\ <br />\end{matrix} \right.\)

1. Sketch the graph of \(y=\mathrm{g}\left( x \right),\) labelling clearly the coordinates of the end-points and the points where the curve crosses the \(x\)-axis, if any.[2]
2. Hence solve the inequality \(1<\mathrm{g}\left( x \right)\le 2\) exactly.[2]
3. Given that \({{\mathrm{g}}^{\mathrm{2}}}\) exists, define \({{\mathrm{g}}^{\mathrm{2}}}\) in a similar form as \(\mathrm{g}\mathrm{.}\)[3]