15 Functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined as follows.
\(\mathrm { f } ( x ) = \sqrt { x }\) for \(x > 0\) and \(\mathrm { g } ( x ) = x ^ { 3 } - x - 6\) for \(x > 2\).
The function \(\mathrm { h } ( x )\) is defined as
\(\mathrm { h } ( x ) = \mathrm { fg } ( x )\).
- Find \(\mathrm { h } ( x )\) in terms of \(x\) and state its domain.
- Find \(\mathrm { h } ( 3 )\).
Fig. 15 shows \(\mathrm { h } ( x )\) and \(\mathrm { h } ^ { - 1 } ( x )\), together with the straight line \(y = x\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-17_780_796_895_242}
\captionsetup{labelformat=empty}
\caption{Fig. 15}
\end{figure} - Determine the gradient of \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\) at the point where \(y = 3\).