2022 ACJC P2 Q1
The function $\text{f}$ is defined by
$\text{f}\left( x \right)=1-ax$, $x\in \mathbb{R}$, where $a$ is a real constant.
The function $\text{g}$ is defined by
$\text{g}\left( x \right)=\left\{ \begin{matrix}
{{x}^{2}}+3\,\,\,\,\,\,\,\text{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\le x\le 2 \\
7-x\,\,\,\,\,\,\,\,\,\,\,\text{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2<x\le 7 \\
\end{matrix} \right.$
(i)
Find the set of possible values of $a$ such that ${{\text{f}}^{-1}}$ exist.
[1]
(ii)
If $a=2$, describe a sequence of transformations that transform the graph of $y=\text{g}\left( x \right)$ onto the graph of $y=\text{fg}\left( x \right)$.
[3]
(iii)
The function $\text{gg}$ is defined by
$\text{gg}\left( x \right)\left\{ \begin{matrix}
{{\text{h}}_{1}}\left( x \right)\,\,\,\,\,\,\,\,\,\,\,\text{for}\,\,\,\,\,\,\,\,\,\,\,\,0\le x\le 2 \\
{{\text{h}}_{2}}\left( x \right)\,\,\,\,\,\,\,\,\,\,\,\text{for}\,\,\,\,\,\,\,\,\,\,\,\,\,2<x<5 \\
{{\text{h}}_{3}}\left( x \right)\,\,\,\,\,\,\,\,\,\,\,\,\text{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\text{5}\le x\le \text{7} \\
\end{matrix} \right.$
Find ${{\text{h}}_{1}}\left( x \right)$, ${{\text{h}}_{2}}\left( x \right)$ and ${{\text{h}}_{3}}\left( x \right)$.
[3]
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