9 Functions f and g are defined by
$$\begin{aligned}
& \mathrm { f } ( x ) = x + \frac { 1 } { x } \quad \text { for } x > 0
& \mathrm {~g} ( x ) = a x + 1 \quad \text { for } x \in \mathbb { R }
\end{aligned}$$
where \(a\) is a constant.
- Find an expression for \(\operatorname { gf } ( x )\).
- Given that \(\operatorname { gf } ( 2 ) = 11\), find the value of \(a\).
- Given that the graph of \(y = \mathrm { f } ( x )\) has a minimum point when \(x = 1\), explain whether or not f has an inverse.
It is given instead that \(a = 5\). - Find and simplify an expression for \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x )\).
- Explain why the composite function fg cannot be formed.
\includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-16_1143_1008_267_566}
The diagram shows a cross-section RASB of the body of an aircraft. The cross-section consists of a sector \(O A R B\) of a circle of radius 2.5 m , with centre \(O\), a sector \(P A S B\) of another circle of radius 2.24 m with centre \(P\) and a quadrilateral \(O A P B\). Angle \(A O B = \frac { 2 } { 3 } \pi\) and angle \(A P B = \frac { 5 } { 6 } \pi\).