2013 HCI C2 BT2 P1 Q11 [Modified]

The functions \(\mathrm{f}\), \(\mathrm{g}\), \(\mathrm{h}\) and \(\mathrm{k}\) are defined by

                                                   \(\mathrm{f}:x\,\mapsto \,{{\mathrm{e}}^{-x}}-\frac{1}{5}\),                    \(x\in \mathbb{R}\);

                                                   \(\mathrm{g}:x\,\mapsto \,\sqrt{\mathrm{f}\left( x \right)}\),                      \(x\in \mathbb{R}\), \(x\le \frac{3}{2}\);

                                                   \(\mathrm{h}:x\,\mapsto \,1-{{x}^{2}}\),                       \(x\in \mathbb{R}\), \(x\ge 0\);

                                                   \(\mathrm{k}:x\,\mapsto \,\frac{2}{x}\),                              \(x\in \mathbb{R}\), \(x>0\).

1. Sketch the graph of \(\mathrm{f}\), showing clearly any points of intersection with the axes and the equations of any asymptotes. Hence find the range of \(\mathrm{f}\).[3]
2. Solve the equation \({{\mathrm{g}}^{-1}}\mathrm{g}\left( x \right)=\mathrm{g}{{\mathrm{g}}^{-1}}\left( x \right)\), giving your answer in exact form.[2]
3. Solve the equation \(\mathrm{h}\left( x \right)={{\mathrm{h}}^{-1}}\left( x \right)\).[2]
4. Show that the function \(\mathrm{gh}\) exists. Hence find the exact range of \(\mathrm{gh}\).[2]
5. Define the function \({{\mathrm{k}}^{2}}\) in a similar form and hence, evaluate \({{\mathrm{k}}^{2013}}\left( 4 \right)\).[3]