The function \(\mathrm{f}\) is defined by \[\mathrm{f}:x\mapsto\frac{2x}{x+1},\qquad x\in\mathbb{R},\quad -1<x\leq0.\]
Find \(\mathrm{f}^{-1}(x)\) and state its domain.[3]
Sketch, on the same diagram, the graphs of \(\mathrm{f}\) and \(\mathrm{f}^{-1}\), showing clearly the relationship between the two graphs and giving the equations of the asymptotes.[3]
The function \(\mathrm{g}\) is defined by \[\mathrm{g}:x\mapsto1-e^x,\qquad x\in\mathbb{R},\quad 0\leq x<\ln3.\]
Explain why \(\mathrm{f}^{-1}\mathrm{g}\) exists.[2]