The function \(\mathrm{f}\) is defined by \(\mathrm{f}:x\mapsto\dfrac{kx+2}{2-x}\), for \(x\in\mathbb{R}\), \(x\ne2\).
If the point with coordinates \((-7,4)\) lies on the curve \(y=\mathrm{f}^{-1}(x)\), show that \(k=3\).[1]
Use \(k=3\) for the rest of this question.
The function \(\mathrm{g}\) is defined by \(\mathrm{g}:x\mapsto\mathrm{f}(|x|)\), for \(x\in\mathbb{R}\), \(x\leq0\) and \(x\ne-2\).
Sketch the graph of \(y=\mathrm{g}(x)\), stating the axial intercept and the equations of asymptotes.[3]
Find \(\mathrm{g}^{-1}(x)\), stating its domain.[4]