The function $\mathrm{f}$ is defined by \(\mathrm{f}(x)=\left\{ \begin{matrix} \frac{x}{2} \qquad

Question

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\left\{ \begin{matrix} \frac{x}{2} \qquad \text{ if}\hspace{0.5em} x\le 0 \ 2\sin x \qquad \text{ if}\hspace{0.5em} 0<x\le 4 \ \end{matrix} \right.$ Sketch the graph of $y=\mathrm{f}(x)$.[2] If the domain of $\mathrm{f}$ is restricted to $x\le k$, state the largest value of $k$, in exact form, for which the function ${{\mathrm{f}}^{-1}}$ exist.[1] Using the domain from part (ii), define ${{\mathrm{f}}^{-1}}$ in a similar form.[4] Solve ${{\mathrm{f}}^{-1}}(x)=\mathrm{f}(x)$.[2] In the rest of the question, the domain of f is as originally defined. The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto -{{x}^{3}},\mathrm{ }x\in \mathbb{R},\mathrm{ }x>0$ Find an expression for $\mathrm{fg}\left( x \right)$.[2]

Accepted Answer

(ii) $k=\frac\pi2$; (iii) $f^{-1}(x)=2x$ for $x\le0$, $f^{-1}(x)=\sin^{-1}(x/2)$ for $0<x\le2$; (iv) $x=0$; (v) $fg(x)=-\frac{x^3}{2}$

2016 PJC P2 Q3

2016 PJC P2 Q3

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