7. The function f is defined by
$$\mathrm { f } ( x ) = 1 - \frac { 2 } { ( x + 4 ) } + \frac { x - 8 } { ( x - 2 ) ( x + 4 ) } , \quad x \in \mathbb { R } , x \neq - 4 , x \neq 2$$
- Show that \(\mathrm { f } ( x ) = \frac { x - 3 } { x - 2 }\)
The function g is defined by
$$\mathrm { g } ( x ) = \frac { \mathrm { e } ^ { x } - 3 } { \mathrm { e } ^ { x } - 2 } , \quad x \in \mathbb { R } , x \neq \ln 2$$
- Differentiate \(\mathrm { g } ( x )\) to show that \(\mathrm { g } ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } }\)
- Find the exact values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) = 1\)