2022 EJC P1 Q4
(a)
The function $\text{f}$ is defined by $\text{f}:x\mapsto {{\left( x+3 \right)}^{2}}-1$ for $x\in \mathbb{R}$, $x\ge -3$.
Find ${{\text{f}}^{-1}}\left( x \right)$, stating the domain of ${{\text{f}}^{-1}}$.
[3]
(b)
The function $\text{g}$ is defined by
$\text{g}\left( x \right)=\left\{ \begin{matrix}
x-1\,\,\,\,\,\,\text{for}\,\,\,\,0<x<1\, \\
{{x}^{2}}-1\,\,\,\,\text{for}\,\,1\le x\le 2 \\
\end{matrix} \right.$
and $\text{g}\left( x \right)=\text{g}\left( x+2 \right)$ for all values of $x$.
Sketch the graph of $\text{g}$ for $-1\le x<3$. Hence state the range of $\text{g}$.
[3]
(c)
The domain of $\text{g}$ is now restricted to $-1\le x<1$.
(i) Explain why the composite function ${{\text{f}}^{-1}}\text{g}$ exists.
[1]
(ii) Find ${{\text{f}}^{-1}}\text{g}\left( x \right)$ in the form
${{\text{f}}^{-1}}\text{g}\left( x \right)=\left\{ \begin{matrix}
\text{p}\left( x \right)\,\,\,\,\text{for}\,\,-1<x\le 0 \\
\text{q}\left( x \right)\,\,\,\,\text{for}\,\,\,\,0<x<1 \\
\end{matrix} \right.$
where $\text{p}\left( x \right)$ and $\text{q}\left( x \right)$ are expressions in terms of $x$ to be determined.
[3]
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