2016 HCI P1 Q11
The functions $\text{f}$ and $\text{g}$ are defined by
$\text{f}:x\mapsto \frac{1}{2}{{e}^{1-{{x}^{2}}}},\text{ }x\in \mathbb{R},\text{ }x\le 1$
$\text{g}:x\mapsto \sqrt{1-\ln x},\text{ }x\in \mathbb{R},\text{ }0<x\le e$.
(i)
Show that $\text{gf}$ exists, and find the range of $\text{gf}$.
[4]
(ii)
Justify, with a reason, whether ${{\text{f}}^{-1}}$ exists.
[2]
(iii)
The domain of $\text{f}$ is restricted to $\left( -\infty ,b \right]$ such that $b$ is the largest value for which the inverse function ${{\text{f}}^{-1}}$ exists. State the value of $b$ and define ${{\text{f}}^{-1}}$clearly.
[4]
(iv)
The graph of $y=\text{h}\left( x \right)$ is obtained by transforming the graph of $y=\text{g}(x)$ in the following $2$ steps.
Step 1: Scale parallel to the $x$-axis by a factor of $2$.
Step 2: Reflect in the $x$-axis.
Define $\text{h}$ in a similar form.
[3]
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