9. \(\quad f ( x ) = 3 - e ^ { 2 x } , \quad x \in \mathbb { R }\).
- State the range of f .
- Find the exact value of \(\mathrm { ff } ( 0 )\).
- Define the inverse function \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
Given that \(\alpha\) is the solution of the equation \(\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )\),
- explain why \(\alpha\) satisfies the equation
$$x = \mathrm { f } ^ { - 1 } ( x )$$
- use the iterative formula
$$x _ { n + 1 } = \mathrm { f } ^ { - 1 } \left( x _ { n } \right)$$
with \(x _ { 0 } = 0.5\) to find \(\alpha\) correct to 3 significant figures.