The one-one function f is defined by \(\mathrm { f } ( x ) = ( x - 3 ) ^ { 2 } - 1\) for \(x < a\), where \(a\) is a constant.
State the greatest possible value of \(a\).
It is given that \(a\) takes this greatest possible value. State the range of f and find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
The function g is defined by \(\mathrm { g } ( x ) = ( x - 3 ) ^ { 2 }\) for \(x \geqslant 0\).
Show that \(\operatorname { gg } ( 2 x )\) can be expressed in the form \(( 2 x - 3 ) ^ { 4 } + b ( 2 x - 3 ) ^ { 2 } + c\), where \(b\) and \(c\) are constants to be found.
Hence expand \(\operatorname { gg } ( 2 x )\) completely, simplifying your answer.
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