2008 HCI P1 Q8
The functions $\text{f}$ and $\text{g}$ are defined by
$\begin{matrix}
& \text{f}:x\mapsto -{{e}^{\left| 3-x \right|}},x\in \mathbb{R}, \\
& \text{g}:x\mapsto a-{{(x-1)}^{2}},x\in \mathbb{R}\text{ and }a\text{ is a constant} \\
\end{matrix}$
Sketch the graph of $y=\text{f}(x)$, and show that $\text{f}$ does not have an inverse.
(i)
The function $\text{f}$ has an inverse if its domain is restricted to $x\ge b$. State the smallest possible value of $b$ and define, in similar form, the inverse function ${{\text{f}}^{-1}}$corresponding to this domain for $\text{f}$.
(ii)
Find the largest value of $a$ such that the composite function ${{\text{f}}^{-1}}\text{g}$ exists and state the range of ${{\text{f}}^{-1}}\text{g}$ for this value of $a$.
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