2025 IGCSE Additional Mathematics May/June 0606/21 Q7

2025 IGCSE Additional Mathematics May/June 0606/21 Q7

8 marks
  1. The function \(\mathrm f\) is defined by \(\mathrm f(x)=2\mathrm e^{-x}+3\) for \(x\in\mathbb R\).
    On the axes, sketch the graph of \(y=\mathrm f(x)\) and hence, on the same axes, sketch the graph of \(y=\mathrm f^{-1}(x)\).
    Show clearly
    • the positions of any points where your graphs meet the coordinate axes
    • the positions of any asymptotes
    [4]
  2. The function \(\mathrm g\) is defined by \(\mathrm g(x)=2-\dfrac3{\mathrm e^x+2}\) for \(x\ge0\).
    Given that \(\mathrm g^{-1}\) exists, find an expression for \(\mathrm g^{-1}(x)\) and state its domain.[4]

Solution:

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Answer:(a) Sketch shown. \(\mathrm f\): intercept \((0,5)\), asymptote \(y=3\); inverse: intercept \((5,0)\), asymptote \(x=3\). (b) \(\mathrm g^{-1}(x)=\ln\left(\dfrac3{2-x}-2\right)\), domain \(1\le x<2\).

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