4.01: Completed square, inverse and composite
The function \(f\) is defined by \(f(x)=2x^2-6x+4\) for all real values of \(x\).
(a)[3]
Write \(f(x)\) in the form \(a(x+b)^2+c\), where \(a\), \(b\) and \(c\) are constants.
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(b)[1]
Hence find the range of \(f\).
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(c)[1]
The function \(g\) is defined by \(g(x)=2x^2-6x+4\) for \(x\leqslant k\), where \(k\) is a constant. Given that \(g\) has an inverse, state the largest possible value of \(k\).
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(d)[3]
The function \(h\) is defined by \(h(x)=\mathrm{e}^{3x}-1\) for \(x\leqslant0\). Given that \(gh(x)\) exists, find \(gh(x)\), simplifying your answer.
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