2022 CJC J1 MYE Q8

The functions \(\mathrm{g}\) and \(\mathrm{h}\) are defined as

\(\mathrm{g}:x\to \left\{ \begin{matrix} 4{{\left( x+1 \right)}^{2}}\mathrm{         for }x\in \mathbb{R}\mathrm{,    }-3\le x<0,  \\ 4x+4\mathrm{             for }x\in \mathbb{R}\mathrm{,       }0\le x<3.  \\ \end{matrix} \right.\)

\(\mathrm{h}:x\mapsto 2\cos x\),          \(x\notin \mathbb{R}\),           \(0\le x\le \frac{\pi }{2}\).

1. Prove that the function \(\mathrm{gh}\) exists and find the range of \(\mathrm{gh}\).[3]

Suppose now that \(\mathrm{g}\left( x \right)=\mathrm{g}\left( x+6 \right)\) for all real values of \(x\).

2. Find the values of \(\mathrm{g}\left( 56 \right)\) and \(\mathrm{g}\left( 99 \right)\).[2]
3. Sketch the graph of \(y=\mathrm{g}\left( x \right)\) for \(-3\le x\le 8\).[3]