The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto x - 2 \mathrm{tan}^{-1}(2x)$, \(

Question

The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto x - 2 \mathrm{tan}^{-1}(2x)$, $-\frac{\sqrt{3}}{2} < x < \frac{\sqrt{3}}{2}$. Show that $\mathrm{f}(x)$ decreases as $x$ increases.[3] Find the range of $\mathrm{f}$.[1] Explain whether the composite function $\mathrm{ff}$ exists.[2] The curve $C$ is obtained by reflecting the graph of $\mathrm{f}$ in the line $y = x$. Write down the equation of $C$.[1] The function $\mathrm{h}$ is defined by $\mathrm{h} : x \mapsto x^2 + 2$, $x \in \mathbb{R}$. The function $\mathrm{g}$ is such that the composite function $\mathrm{gh}$ exists and $$\mathrm{gh}(x) = x^4 - 2x^2 + 5, \quad x \in \mathbb{R}.$$ Find an expression for $\mathrm{g}(x)$.[2]

Accepted Answer

(a)(i) $f'(x)<0$ on its domain. (ii) $R_f=\left(\dfrac{\sqrt3}{2}-\dfrac{2\pi}{3},\dfrac{2\pi}{3}-\dfrac{\sqrt3}{2}\right)$ (iii) $ff$ does not exist. (iv) $x=y-2\tan^{-1}(2y)$, $-\dfrac{\sqrt3}{2}<y<\dfrac{\sqrt3}{2}$. (b) $g(x)=x^2-6x+13$ for $x\ge2$.

2009 IJC P1 Q10

2009 IJC P1 Q10

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