6. The functions \(f\) and \(g\) are defined by
$$\begin{aligned}
& \mathrm { f } : x \mapsto \mathrm { e } ^ { x } + 2 , \quad x \in \mathbb { R }
& \mathrm {~g} : x \mapsto \ln x , \quad x > 0
\end{aligned}$$
- State the range of f.
- Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
- Find the exact value of \(x\) for which \(\mathrm { f } ( 2 x + 3 ) = 6\)
- Find \(\mathrm { f } ^ { - 1 }\), the inverse function of f , stating its domain.
- On the same axes sketch the curves with equation \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), giving the coordinates of all the points where the curves cross the axes.