The function $\mathrm{f}$ is defined by \({\mathrm{f}\left( x \right)=\left\{ \begin{matrix} 2-\ln

Question

The function $\mathrm{f}$ is defined by ${\mathrm{f}\left( x \right)=\left\{ \begin{matrix} 2-\ln \left( x+\mathrm{e} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for }\,\,\,\,\,\,-\mathrm{e} < x < 0 \ \cos \left( \frac{\pi }{2}x \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for }\,\,\,\,\,\,\,\,0\le x\le 1 \ \end{matrix} \right.}$. Given that the inverse function ${{\mathrm{f}}^{-1}}$ exists, find the exact value of $m$ if ${{\mathrm{f}}^{-1}}\left( m \right)=-1$.[2] Another function $\mathrm{g}$ is defined by $\mathrm{g}\left( x \right)=\sin x$, $\qquad 0\le x\le \frac{\pi }{4}$. Explain why the composite function $\mathrm{fg}$ exists.[2] Find the exact value of $\mathrm{fg}\left( \frac{\pi }{6} \right)$.[2] The domain of $\mathrm{f}$ is further restricted to $0\le x\le 1$. By considering ${{\mathrm{f}}^{2}}\left( 1 \right)$ and ${{\mathrm{f}}^{3}}\left( 1 \right)$, deduce the value(s) of ${{\mathrm{f}}^{n}}\left( 1 \right)$ where $n$ is a positive integer.[2]

Accepted Answer

(a) ${2 - \ln(\mathrm{e} - 1)}$ (b) Since ${\mathrm{R}_{\mathrm{g}} \subseteq \mathrm{D}_{\mathrm{f}}}$. Hence, ${\mathrm{f}\mathrm{g}}$ exists. (c) ${= \frac{1}{\sqrt{2}}}$ (d) ${\mathrm{f}^n(1) = \begin{cases} 0 & \text{if } n \text{ is odd} \ 1 & \text{if } n \text{ is even} \end{cases}}$

2024 HCI Promo Q8

2024 HCI Promo Q8

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