Express \(2 x ^ { 2 } + 12 x + 11\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 12 x + 11\) for \(x \leqslant - 4\).
Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
The function g is defined by \(\mathrm { g } ( x ) = 2 x - 3\) for \(x \leqslant k\).
For the case where \(k = - 1\), solve the equation \(\operatorname { fg } ( x ) = 193\).
State the largest value of \(k\) possible for the composition fg to be defined.