2019 JPJC P1 Q7

It is given that

\({\mathrm{f}(x) = \begin{cases} 4a^2 - x^2, & \mathrm{for } \ 0 < x \leq 2a, \\ 2a(x - 2a), & \mathrm{for } \ 2a < x \leq 4a, \end{cases}}\)

and that \(\mathrm{f}(x)=\mathrm{f}(x+4a)\) for all real values of \(x\), where \(a\) is a positive real constant.

1. Evaluate \(\mathrm{f}(2019a)\) in terms of \(a\).[1]
2. Sketch the graph of \(y=\mathrm{f}(x)\) for \(-3a\le x\le 5a\).[3]

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

\(\mathrm{g}:x\mapsto \sqrt{4{{a}^{2}}-{{(x-2a)}^{2}}}\), \({2a < x < 4a}\).

3. Determine whether the composite function \(\mathrm{gf}\) exists, justifying your answer.[1]
4. Give, in terms of \(a\), a definition of \(\mathrm{fg}\).[2]
5. Given that \({{\left( \mathrm{fg} \right)}^{-1}}\left( 27 \right)=\frac{7}{2}a\), find the exact value of \(a\).[2]