2008 YJC P1 Q6 [Modified]

The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined as follows:

\(\mathrm{f}:x\mapsto {{\left( x-2 \right)}^{2}}-3\), \(x\le 2\)

\(\mathrm{g}:x\mapsto 4\cos x\), \(0\le x<\pi \)

1. Find \({{\mathrm{f}}^{-1}}\) in a similar form.[3]
2. Sketch on the same diagram, the graphs of \(y=\mathrm{f}\left( x \right)\) and \(y={{\mathrm{f}}^{-1}}\left( x \right)\).
   Hence, solve the equation \({{\mathrm{f}}^{-1}}\left( x \right)=\mathrm{f}\left( x \right)\), leaving your answer in exact form.[4]
3. Without finding \({{\mathrm{g}}^{-1}}\), solve \({{\mathrm{g}}^{-1}}\left( x \right)=\frac{\pi }{4}\), giving your answer in exact form.[2]