2023 DHS P1 Q11

The function \(\mathrm{f}\) is given by

\(\mathrm{f}:x\mapsto a+\frac{3a}{\left( x+3 \right)\left( x-1 \right)}\), where \(a > 0\).

1. Using calculus, find the \(x\)-coordinate of the stationary point of \(y=\mathrm{f}\left( x \right)\).[2]
2. Sketch the graph of \(y=\mathrm{f}\left( x \right)\), clearly stating the coordinates of the stationary point and equations of any asymptotes.[3]
3. State the range of values of \(x\) for which the graph of \(y=\mathrm{f}\left( x \right)\) is increasing and concave downwards.[1]
4. Sketch the graph of \(y=\mathrm{f}\,\mathrm{ }\!\!'\!\!\mathrm{ }\left( x \right)\), clearly stating the coordinates of the points where the graph crosses the axes and equations of any asymptotes. Leave your answers in terms of \(a\) where appropriate.[3]

It is given that \(\mathrm{g}\left( x \right)=\mathrm{f}\left( x \right)\) for \(x\ge 2\). Suppose the composite function \(\mathrm{gg}\) exists.

5. Deduce, with a reason, the condition on \(a\).[3]
6. Find the exact range of \(\mathrm{gg}\) for \(a=5\).[2]