2013 HCI C2 BT2 P1 Q11 [Modified]
The functions $\text{f}$, $\text{g}$, $\text{h}$ and $\text{k}$ are defined by
$\text{f}:x\,\mapsto \,{{\text{e}}^{-x}}-\frac{1}{5}$, $x\in \mathbb{R}$;
$\text{g}:x\,\mapsto \,\sqrt{\text{f}\left( x \right)}$, $x\in \mathbb{R}$, $x\le \frac{3}{2}$;
$\text{h}:x\,\mapsto \,1-{{x}^{2}}$, $x\in \mathbb{R}$, $x\ge 0$;
$\text{k}:x\,\mapsto \,\frac{2}{x}$, $x\in \mathbb{R}$, $x>0$.
(i)
Sketch the graph of $\text{f}$, showing clearly any points of intersection with the axes and the equations of any asymptotes. Hence find the range of $\text{f}$.
[3]
(ii)
Solve the equation ${{\text{g}}^{-1}}\text{g}\left( x \right)=\text{g}{{\text{g}}^{-1}}\left( x \right)$, giving your answer in exact form.
[2]
(iii)
Solve the equation $\text{h}\left( x \right)={{\text{h}}^{-1}}\left( x \right)$.
[2]
(iv)
Show that the function $\text{gh}$ exists. Hence find the exact range of $\text{gh}$.
[2]
(v)
Define the function ${{\text{k}}^{2}}$ in a similar form and hence, evaluate ${{\text{k}}^{2013}}\left( 4 \right)$.
[3]
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