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2021 IGCSE Additional Mathematics May/June 0606/23 Q9

2021 IGCSE Additional Mathematics May/June 0606/23 Q9

10 marks
  1. The function \(\mathrm{f}\) is defined, for all real \(x\), by \(\mathrm{f}(x)=13-4x-2x^2\).
    1. Write \(\mathrm{f}(x)\) in the form \(a+b(x+c)^2\), where \(a\), \(b\) and \(c\) are constants.[3]
    2. Hence write down the range of \(\mathrm{f}\).[1]
  2. The function \(\mathrm{g}\) is defined, for \(x\ge1\), by \(\mathrm{g}(x)=\sqrt{x^2+2x-1}\).
    1. Given that \(\mathrm{g}^{-1}(x)\) exists, write down the domain and range of \(\mathrm{g}^{-1}\).[2]
    2. Show that \(\mathrm{g}^{-1}(x)=-1+\sqrt{px^2+q}\), where \(p\) and \(q\) are integers.[4]

Solution:

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Answer:(a)(i) \(\mathrm{f}(x)=15-2(x+1)^2\) (ii) \(\mathrm{f}(x)\le15\) (b)(i) Domain \(x\ge\sqrt{2}\), range \(\mathrm{g}^{-1}(x)\ge1\) (ii) \(\mathrm{g}^{-1}(x)=-1+\sqrt{x^2+2}\), so \(p=1\), \(q=2\)

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