The function $\mathrm{f}$ is defined by \[\mathrm{f} : x \mapsto \frac{ax}{bx - a}, \quad \text{fo

Question

The function $\mathrm{f}$ is defined by $$\mathrm{f} : x \mapsto \frac{ax}{bx - a}, \quad \text{for}\hspace{0.5em} x \in \mathbb{R}, \ x \neq \frac{a}{b},$$ where $a$ and $b$ are non-zero constants. Find $\mathrm{f}^{-1}(x)$. Hence or otherwise find $\mathrm{f}^2(x)$ and state the range of $\mathrm{f}^2$.[5] The function $\mathrm{g}$ is defined by $\mathrm{g} : x \mapsto \frac{1}{x}$ for all real non-zero $x$. State whether the composite function $\mathrm{fg}$ exists, justifying your answer.[2] Solve the equation $\mathrm{f}^{-1}(x) = x$.[3]

Accepted Answer

(i) $\mathrm{f}^{-1}(x)=\frac{ax}{bx-a}$, $\mathrm{f}^2(x)=x$, range $\mathbb{R}\setminus\left\{\frac{a}{b}\right\}$ (ii) $\mathrm{fg}$ does not exist (iii) $x=0$ or $x=\frac{2a}{b}$

N2009 P2 Q3

N2009 P2 Q3

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