11 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 6 x + 5\) for \(x \in \mathbb { R }\).
- Find the set of values of \(p\) for which the equation \(\mathrm { f } ( x ) = p\) has no real roots.
The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 6 x + 5\) for \(0 \leqslant x \leqslant 4\).
- Express \(\mathrm { g } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
- Find the range of g .
The function h is defined by \(\mathrm { h } : x \mapsto 2 x ^ { 2 } - 6 x + 5\) for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.
- State the smallest value of \(k\) for which h has an inverse.
- For this value of \(k\), find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).