The function $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto 6+\lambda x-{{x}^{2}}

Question

The function $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto 6+\lambda x-{{x}^{2}}$, $x\in \mathbb{R}$, $\mathrm{g}:x\mapsto 1+7{{\mathrm{e}}^{-x}}$, $x\ge 0$. For this part of the question, it is given that $\lambda =5$. Show that the composite function $\mathrm{fg}$ exists. Hence find the range of $\mathrm{fg}$. [3] It is not given that $\lambda $ is a real constant that is not necessary equals to $5$. If the domain of $\mathrm{f}$ is restricted to $x\le \frac{\lambda }{2}$, find ${{\mathrm{f}}^{-1}}$ in a similar form. [4]

Accepted Answer

(i) $fg$ exists, $R_{fg} = \left[-18, \frac{49}{4}\right]$ (ii) $f^{-1}(x) = \frac{\lambda}{2} - \sqrt{6 + \frac{\lambda^2}{4} - x}$, $x \leq 6 + \frac{\lambda^2}{4}$

2021 ASRJC Promo Q6

2021 ASRJC Promo Q6

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