2016 PJC P2 Q3
The function $\text{f}$ is defined by
$\text{f}(x)=\left\{ \begin{matrix}
& \frac{x}{2}\text{ if }x\le 0 \\
& 2\sin x\text{ if }0<x\le 4 \\
\end{matrix} \right.$
(i)
Sketch the graph of $y=\text{f}(x)$.
[2]
(ii)
If the domain of $\text{f}$ is restricted to $x\le k$, state the largest value of $k$, in exact form, for which the function ${{\text{f}}^{-1}}$ exist.
[1]
(iii)
Using the domain from part (ii), define ${{\text{f}}^{-1}}$ in a similar form.
[4]
(iv)
Solve ${{\text{f}}^{-1}}(x)=\text{f}(x)$.
[2]
In the rest of the question, the domain of f is as originally defined.
The function $\text{g}$ is defined by $\text{g}:x\mapsto -{{x}^{3}},\text{ }x\in \mathbb{R},\text{ }x>0$
(v)
Find an expression for $\text{fg}\left( x \right)$.
[2]
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