4. The function \(f\) is defined by
$$f : x \mapsto \frac { 2 ( x - 1 ) } { x ^ { 2 } - 2 x - 3 } - \frac { 1 } { x - 3 } , \quad x > 3$$
- Show that \(\mathrm { f } ( x ) = \frac { 1 } { x + 1 } , \quad x > 3\).
- Find the range of f.
- Find \(\mathrm { f } ^ { - 1 } ( x )\). State the domain of this inverse function.
The function \(g\) is defined by
$$\mathrm { g } : x \mapsto 2 x ^ { 2 } - 3 , \quad x \in \mathbb { R }$$
- Solve \(\mathrm { fg } ( x ) = \frac { 1 } { 8 }\).