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

(iii)$D_g^{-21}=R_g=(-\infty,1)\cup(1,\infty)$ (v)$\pm\sqrt3$

2025 MI P2 Q5

2025 MI P2 Q5