11 The function f is defined by \(\mathrm { f } ( x ) = 3 + 6 x - 2 x ^ { 2 }\) for \(x \in \mathbb { R }\).
- Express \(\mathrm { f } ( x )\) in the form \(a - b ( x - c ) ^ { 2 }\), where \(a , b\) and \(c\) are constants, and state the range of f .
- The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = \mathrm { h } ( x )\) by a reflection in one of the axes followed by a translation. It is given that the graph of \(y = \mathrm { h } ( x )\) has a minimum point at the origin.
Give details of the reflection and translation involved.
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The function g is defined by \(\mathrm { g } ( x ) = 3 + 6 x - 2 x ^ { 2 }\) for \(x \leqslant 0\). - Sketch the graph of \(y = \mathrm { g } ( x )\) and explain why g is a one-one function. You are not required to find the coordinates of any intersections with the axes.
- Sketch the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\) on your diagram in (c), and find an expression for \(\mathrm { g } ^ { - 1 } ( x )\). You should label the two graphs in your diagram appropriately and show any relevant mirror line.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
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