8. The function \(f\) is defined by
$$\mathrm { f } : x \rightarrow 3 - 2 \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$
- Find the inverse function, \(\mathrm { f } ^ { - 1 } ( x )\) and give its domain.
- Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \ln x\).
The equation \(\mathrm { f } ( t ) = k \mathrm { e } ^ { t }\), where \(k\) is a positive constant, has exactly one real solution.
- Find the value of \(k\).