2021 JPJC P1 Q6

It is given that

\(\mathrm{f}:x\mapsto \left\{ \begin{matrix} -a\sqrt{1-\frac{{{x}^{2}}}{4},}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-2x\le x<2 \\ 3x-6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x\le x<3 \\ \end{matrix} \right.\)

and that \(\mathrm{f}\left( x \right)=\mathrm{f}\left( x+5 \right)\) for all real values of \(x\), where \(a\) is a positive constant.

1. Find \(\mathrm{f}\left( 2021 \right)\).[2]
2. Sketch the graph of \(y=\mathrm{f}\left( x \right)\) for \(-5\le x\le 7\), stating the coordinates of the maximum points, end-points and points where the curve crosses the \(x\)- and \(y\)-axes.[4]
3. On a separate diagram, sketch the curve with equation \(y=3\mathrm{f}\left( x+2 \right)\) for \(-2\le x\le 3\), stating the coordinates of the points where the curve crosses the \(x\)- and \(y\)-axes.[2]
4. Another function \(\mathrm{g}\) is defined by
   \(\mathrm{g}:x\mapsto \sqrt{x}+2\), for \(0\le x<1\).
   State the range of \(\mathrm{g}\) and find \(\mathrm{fg}\left( x \right)\), stating its domain.[3]