5.
\includegraphics[max width=\textwidth, alt={}, center]{3e18cb7c-1a67-4152-8628-76847e368882-4_639_1177_264_445}
Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where
$$f ( x ) = \frac { x ^ { 2 } + 16 } { 3 x } \quad x \neq 0$$
The curve has a maximum at the point \(A\) with coordinates \(( a , b )\).
- Find the value of \(a\) and the value of \(b\).
The function g is defined as
$$\mathrm { g } : x \mapsto \frac { x ^ { 2 } + 16 } { 3 x } \quad a \leqslant x < 0$$
where \(a\) is the value found in part (a).
- Write down the range of g .
- On the same axes sketch \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\).
- Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { g } ^ { - 1 }\)
- Solve the equation \(\mathrm { g } ( x ) = \mathrm { g } ^ { - 1 } ( x )\).