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2021 IGCSE Additional Mathematics May/June 0606/22 Q13

2021 IGCSE Additional Mathematics May/June 0606/22 Q13

7 marks

The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined, for \(x>0\), by

\(\mathrm{f}(x)=\dfrac{2x^2-1}{3x},\)

\(\mathrm{g}(x)=\dfrac1x.\)

  1. Find and simplify an expression for \(\mathrm{fg}(x)\).[2]
    1. Given that \(\mathrm{f}^{-1}\) exists, write down the range of \(\mathrm{f}^{-1}\).[1]
    2. Show that \(\mathrm{f}^{-1}(x)=\dfrac{px+\sqrt{qx^2+r}}{4}\), where \(p\), \(q\) and \(r\) are integers.[4]

Solution:

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Answer:(a) \(\mathrm{fg}(x)=\dfrac{2-x^2}{3x}\) (b)(i) \(\mathrm{f}^{-1}(x)>0\) (ii) \(\mathrm{f}^{-1}(x)=\dfrac{3x+\sqrt{9x^2+8}}{4}\), so \(p=3\), \(q=9\), \(r=8\)

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