10 The function f is such that \(\mathrm { f } ( x ) = 2 x + 3\) for \(x \geqslant 0\). The function g is such that \(\mathrm { g } ( x ) = a x ^ { 2 } + b\) for \(x \leqslant q\), where \(a , b\) and \(q\) are constants. The function fg is such that \(\operatorname { fg } ( x ) = 6 x ^ { 2 } - 21\) for \(x \leqslant q\).
- Find the values of \(a\) and \(b\).
- Find the greatest possible value of \(q\).
It is now given that \(q = - 3\).
- Find the range of fg.
- Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and state the domain of \(( \mathrm { fg } ) ^ { - 1 }\).