The function $\mathrm{f}$ is defined by \[\mathrm{f} : x \mapsto \frac{x-1}{x}, x \in \mathbb{R},

Question

The function $\mathrm{f}$ is defined by $$\mathrm{f} : x \mapsto \frac{x-1}{x}, x \in \mathbb{R}, x \neq 0, x \neq 1$$ Show that $\mathrm{f}^2(x) = \mathrm{f}^{-1}(x)$.[3] Find $\mathrm{f}^3(x)$ in simplified form.[1] Find $\mathrm{f}^{2030}(5)$.[2] Functions $\mathrm{g}$ and $\mathrm{h}$ are defined by $$\mathrm{g} : x \mapsto \frac{x-1}{x}, x \in \mathbb{R}, x \geq 1$$ $$\mathrm{h} : x \mapsto -\sin ax, x \in \mathbb{R}$$ where $a$ is a positive constant. Find the value of $a$ given that the range of $\mathrm{hg}$ is $(-1, 0]$.[2]

Accepted Answer

(a) $\mathrm f^{-1}(x)=\mathrm f^2(x)=\dfrac{1}{1-x}$ (b) $\mathrm f^3(x)=x$ (c) $-\dfrac14$ (d) $a=\dfrac\pi2$

2025 VJC P1 Q2

2025 VJC P1 Q2

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