2021 ACJC Promo Q7

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

\(\mathrm{f}:x\mapsto \frac{1}{\left| 1-{{x}^{2}} \right|}\), \(x\in \mathbb{R}\), \(-2\le x<-1\),

\(\mathrm{g}:x\mapsto -{{\left( x-2 \right)}^{2}}+k\), \(x\in \mathbb{R}\), \(x\ge 0\) where \(k\) is a constant.

1. Sketch on the same diagram the graphs of
   1. \(y=\mathrm{f}\left( x \right)\)
   2. \(y={{\mathrm{f}}^{-1}}\left( x \right)\)
   3. \(y={{\mathrm{f}}^{-1}}\mathrm{f}\left( x \right)\)
   stating the equations of any asymptotes and the coordinates of any endpoints.[3]
2. Find \({{\mathrm{f}}^{-1}}\) and state the domain of \({{\mathrm{f}}^{-1}}\).[3]
3. Show that the composite function \(\mathrm{gf}\) exists and find its range.[2]