A function \(\mathrm{f}\) is such that \(\mathrm{f}(x)=2+\mathrm{e}^{-3x}\), \(x\in\mathbb R\).
Write down the range of \(\mathrm{f}\).[1]
Find an expression for \(\mathrm{f}^{-1}\).[2]
On the axes, sketch the graphs of \(y=\mathrm{f}(x)\) and \(y=\mathrm{f}^{-1}(x)\), stating the coordinates of the points where the curves meet the coordinate axes. State the equations of any asymptotes. Label your curves.[4]
A function \(\mathrm{g}\) is such that \(\mathrm{g}(x)=x^{\frac32}+4\), \(x\geq0\).
Find the exact solution of the equation \(\mathrm{gf}(x)=12\).[4]