ACJC/2021/JC1/Promo/P1/Q07

The functions $f$ and $g$ are defined by
$\mathrm{f}: x \mapsto \frac{1}{\left|1-x^2\right|}, x \in \mathbb{R},-2 \leq x<-1,$
g: $x \mapsto-(x-2)^2+k, x \in \mathbb{R}, \quad x \geq 0$ where $k$ is a constant.

(i) Sketch on the same diagram the graphs of
(a) $y=\mathrm{f}(x)$
(b) $y=\mathrm{f}^{-1}(x)$
(c) $y=\mathrm{f}^{-1} \mathrm{f}(x)$

stating the equations of any asymptotes and the coordinates of any endpoints.

[3]

(ii) Find $\mathrm{f}^{-1}$ and state the domain of $\mathrm{f}^{-1}$.

[3]

(iii) Show that the composite function gf exists and find its range.

[2]