2024 DHS P2 Q4

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

\(\mathrm{f}\left( x \right)=\left\{ \begin{matrix} 2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\mathrm{ }1\le x<8, \\ \frac{x}{2}+12\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\mathrm{ }8\le x\le 30. \\ \end{matrix} \right.\)

1. Sketch the graph of \(y=\mathrm{f}\left( x \right)\).[1]
2. Solve the equation \(\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)\).[2]
3. Show that the composite function \({{\mathrm{f}}^{2}}\) exists and find its range.[2]
4. Given that the composite function \({{\mathrm{f}}^{n}}\) exists for \(n\ge 3\), find the range of \({{\mathrm{f}}^{3}}\), \({{\mathrm{f}}^{4}}\) and hence find the range of \({{\mathrm{f}}^{n}}\) as \(n\to \infty \).[3]