The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \sqrt{5 + 2x}$, for \(x \in

Question

The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \sqrt{5 + 2x}$, for $x \in \mathbb{R}$, $x > -\frac{5}{2}$. Find the value of $a$ such that $\mathrm{f}^2(a) = 5$.[2] Another function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto \left|\frac{x}{1 - 2x}\right|$, for $x \in \mathbb{R}$, $x \le k$. State the largest value of $k$ for $\mathrm{g}^{-1}$ to exist.[1] It is now given that $k = -1$. Find $\mathrm{g}^{-1}(x)$ and state its domain.[3] Explain why the composite function $\mathrm{fg}$ exists. State the exact range of $\mathrm{fg}$.[2]

Accepted Answer

(a) $a=\frac{95}{2}$ (b) $k=0$ (c) $g^{-1}(x)=\frac{x}{2x-1}$, with domain $\left[\frac13,\frac12\right)$ (d) $fg$ exists, and $\operatorname{range}(fg)=\left[\sqrt{\frac{17}{3}},\sqrt6\right)$

2024 ASRJC Promo Q7

2024 ASRJC Promo Q7

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