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2020 IGCSE Additional Mathematics February/March 0606/22 Q10

2020 IGCSE Additional Mathematics February/March 0606/22 Q10

9 marks
  1. \(\mathrm{g}(x)=3+\dfrac{1}{x}\) for \(x\ge1\).
    1. Find an expression for \(\mathrm{g}^{-1}(x)\).[2]
    2. Write down the range of \(\mathrm{g}^{-1}\).[1]
    3. Find the domain of \(\mathrm{g}^{-1}\).[2]
  2. \(\mathrm{h}(x)=2\ln(3x-1)\) for \(x\ge\dfrac23\).
    The graph of \(y=\mathrm{h}(x)\) intersects the line \(y=x\) at two distinct points. On the axes below, sketch the graph of \(y=\mathrm{h}(x)\) and hence sketch the graph of \(y=\mathrm{h}^{-1}(x)\).[4]

Solution:

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Answer:(a)(i) \(\mathrm{g}^{-1}(x)=\dfrac{1}{x-3}\) (ii) \(\mathrm{g}^{-1}(x)\ge1\) (iii) \(3<x\le4\) (b) Sketch shown: \(y=\mathrm{h}(x)\) from \(\left(\dfrac23,0\right)\) and its reflection \(y=\mathrm{h}^{-1}(x)\) in \(y=x\) from \(\left(0,\dfrac23\right)\), meeting twice on \(y=x\)

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