Express \(x ^ { 2 } - 2 x - 15\) in the form \(( x + a ) ^ { 2 } + b\).
The function f is defined for \(p \leqslant x \leqslant q\), where \(p\) and \(q\) are positive constants, by
$$f : x \mapsto x ^ { 2 } - 2 x - 15$$
The range of f is given by \(c \leqslant \mathrm { f } ( x ) \leqslant d\), where \(c\) and \(d\) are constants.
State the smallest possible value of \(c\).
For the case where \(c = 9\) and \(d = 65\),
find \(p\) and \(q\),
find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).