The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows: \[\mathrm{f} : x \mapsto \fr

Question

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows: $$\mathrm{f} : x \mapsto \frac{\ln(3x - 2)}{x}, \, x > \frac{2}{3}$$ $$\mathrm{g} : x \mapsto 3x^2 - 12x + 13, \, x \in \mathbb{R}$$ Sketch the graph of $y = \mathrm{f}(x)$, indicating the equations of asymptotes; coordinates of intersections with the axes and turning points, if any.[3] Show that $\mathrm{fg}$ exists and find its range.[2] The function $\mathrm{h}$ is defined as $$\mathrm{h} : x \mapsto \mathrm{g}(x), \, x \le k$$ such that $\mathrm{h}^{-1}$ exists. State the largest value of $k$. For this value of $k$, find $\mathrm{h}^{-1}$ in a similar form.[4]

Accepted Answer

(i) Asymptotes $x=\dfrac23$ and $y=0$, $x$-intercept $(1,0)$, no $y$-intercept, maximum $(2.11,0.695)$ (ii) $R_{\mathrm{fg}}=[0,0.695]$ (iii) $k=2$, $\mathrm h^{-1}(x)=2-\sqrt{\dfrac{x-1}{3}}$ for $x\ge1$

2024 ACJC Promo Q5

2024 ACJC Promo Q5

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