The functions f and g are defined for \(x \geqslant 0\) by
$$\begin{aligned}
& \mathrm { f } : x \mapsto ( a x + b ) ^ { \frac { 1 } { 3 } } , \text { where } a \text { and } b \text { are positive constants, }
& \mathrm { g } : x \mapsto x ^ { 2 } .
\end{aligned}$$
Given that \(\mathrm { fg } ( 1 ) = 2\) and \(\operatorname { gf } ( 9 ) = 16\),
calculate the values of \(a\) and \(b\),
obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
A point \(P\) travels along the curve \(y = \left( 7 x ^ { 2 } + 1 \right) ^ { \frac { 1 } { 3 } }\) in such a way that the \(x\)-coordinate of \(P\) at time \(t\) minutes is increasing at a constant rate of 8 units per minute. Find the rate of increase of the \(y\)-coordinate of \(P\) at the instant when \(P\) is at the point \(( 3,4 )\).