9 Functions f and g are defined by
$$\begin{aligned}
& \mathrm { f } ( x ) = 2 x ^ { 2 } + 8 x + 1 \quad \text { for } x \in \mathbb { R }
& \mathrm {~g} ( x ) = 2 x - k \quad \text { for } x \in \mathbb { R }
\end{aligned}$$
where \(k\) is a constant.
- Find the value of \(k\) for which the line \(y = \mathrm { g } ( x )\) is a tangent to the curve \(y = \mathrm { f } ( x )\).
- In the case where \(k = - 9\), find the set of values of \(x\) for which \(\mathrm { f } ( x ) < \mathrm { g } ( x )\).
- In the case where \(k = - 1\), find \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x )\) and solve the equation \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x ) = 0\).
- Express \(\mathrm { f } ( x )\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants, and hence state the least value of \(\mathrm { f } ( x )\).