2021 ACJC Promo Q7
The functions $\text{f}$ and $\text{g}$ are defined by
$\text{f}:x\mapsto \frac{1}{\left| 1-{{x}^{2}} \right|}$, $x\in \mathbb{R}$, $-2\le x<-1$,
$\text{g}:x\mapsto -{{\left( x-2 \right)}^{2}}+k$, $x\in \mathbb{R}$, $x\ge 0$ where $k$ is a constant.
(i)
Sketch on the same diagram the graphs of
(a) $y=\text{f}\left( x \right)$
(b) $y={{\text{f}}^{-1}}\left( x \right)$
(c) $y={{\text{f}}^{-1}}\text{f}\left( x \right)$
stating the equations of any asymptotes and the coordinates of any endpoints.
[3]
(ii)
Find ${{\text{f}}^{-1}}$ and state the domain of ${{\text{f}}^{-1}}$.
[3]
(iii)
Show that the composite function $\text{gf}$ exists and find its range.
[2]
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