The function $\mathrm{g}$ is given by $\mathrm{g} : x \mapsto \frac{x-3}{x-1}$, for \(x \in \mat

Question

The function $\mathrm{g}$ is given by $\mathrm{g} : x \mapsto \frac{x-3}{x-1}$, for $x \in \mathbb{R}$, $x \neq 1$. Sketch the graph of $y = \mathrm{g}(x)$, stating clearly the coordinates of any points of intersection with the axes and the equation of any asymptotes.[3] Explain if $\mathrm{g}^{-1}$ exists.[1] Find $\mathrm{g}^{-1}(x)$ and state the domain of $\mathrm{g}^{-1}$.[3] A function $\mathrm{f}$ is said to be self-inverse if $\mathrm{f}(x) = \mathrm{f}^{-1}(x)$ for all $x \in \mathbb{R}$ in the domain of $\mathrm{f}$. Show that $\mathrm{g}$ is self-inverse.[1] Without the use of a calculator, solve the equation $\mathrm{g}^2(x) = -\mathrm{g}^{-1}(x)$.[2]

Accepted Answer

(i) Intercepts $(0,3)$, $(3,0)$; asymptotes $x=1$, $y=1$; sketch shown. (ii) Inverse exists. (iii) $\mathrm{g}^{-1}(x)=\frac{x-3}{x-1}$, $x\ne1$. (iv) Self-inverse, as shown. (v) $x=\pm\sqrt3$.

2025 MI P2 Q5

2025 MI P2 Q5

Solution:

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