The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \left| 3{{\mathrm{e}

Question

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \left| 3{{\mathrm{e}}^{-x}}-8 \right|-2$, $x\in \mathbb{R}$, $x\ge \lambda $, $\mathrm{g}:x\mapsto a\sin x$, $x\in \mathbb{R}$, $0 where $a$ is a positive constant. Find the smallest value of $\lambda $, in exact form, for ${{\mathrm{f}}^{-1}}$ to exist. [2] For the rest of the question, $\lambda $ takes the value found in part (i). Find ${{\mathrm{f}}^{-1}}\left( x \right)$. [3] Find, in terms of $a$, the range of $\mathrm{fg}$. [2]

Accepted Answer

(i) $\lambda =-\ln \frac{8}{3}$ (ii) $-\ln \left( 2-\frac{x}{3} \right)$ (iii) $(3,6-3{{\mathrm{e}}^{-a}}]$

2020 VJC Promo Q6

2020 VJC Promo Q6

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