The function \(\mathrm{f}\) is defined by
where \(a\) is a positive constant.
Sketch the curve \(y=\mathrm{f}(x)\), stating the equation of the asymptote and the coordinates of the endpoint. Explain, stating the reason clearly, why \(\mathrm{f}\) has an inverse.[3]
Find \(\mathrm{f}^{-1}(x)\) and state its domain.[3]
Find the range of values of \(x\) in terms of \(a\), for which \(\mathrm{f}(x)=\mathrm{f}^{-1}(x)\).[1]
For positive integers \(k\), find \(\mathrm{f}^{2k+1}\left(-\dfrac12a\right)\) in terms of \(a\).[2]
Given that \(\mathrm{f}\mathrm{g}(x)=x+a\) for \(x\in\mathbb{R}\), \(x\leq-a\), find \(\mathrm{g}(x)\), stating its domain.[2]