(a(i))[4]
For $x \geq 0$, the functions are given by $f: x \mapsto (ax + b)^{\frac{1}{3}}$, where $a$ and $b$ are positive constants, and $g: x \mapsto x^2$. If $fg(1) = 2$ and $fg(9) = 16$, calculate the values of $a$ and $b$.
(a(ii))[4]
Find an expression for $f^{-1}(x)$ and state the domain of $f^{-1}$.
(b)[5]
A point $P$ moves along the curve $y = (7x^2 + 1)^{\frac{1}{3}}$ so that, after $t$ minutes, the $x$-coordinate of $P$ is rising at a steady rate of $8$ units per minute. Find the rate of increase of the $y$-coordinate of $P$ at the moment when $P$ is at the point $(3, 4)$.