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2022 IGCSE Additional Mathematics February/March 0606/12 Q9

2022 IGCSE Additional Mathematics February/March 0606/12 Q9

12 marks
  1. The function \(\mathrm{f}\) is such that \(\mathrm{f}(x)=\ln(5x+2)\), for \(x>a\), where \(a\) is as small as possible.
    1. Write down the value of \(a\).[1]
    2. Hence find the range of \(\mathrm{f}\).[1]
    3. Find \(\mathrm{f}^{-1}(x)\), stating its domain.[3]
    4. On the axes, sketch the graphs of \(y=\mathrm{f}(x)\) and \(y=\mathrm{f}^{-1}(x)\), stating the exact values of the intercepts of the curves with the coordinate axes.[4]
  2. The function \(\mathrm{g}\) is such that \(\mathrm{g}:x\mapsto x^{\frac12}-4\), for \(x>0\). Solve the equation \(\mathrm{g}^2(x)=-2\).[3]

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Answer:(a)(i) \(a=-0.4\) (ii) \(\mathrm{f}(x)\in\mathbb{R}\) (iii) \(\mathrm{f}^{-1}(x)=\dfrac{\mathrm{e}^x-2}{5}\), domain \(x\in\mathbb{R}\) (iv) Sketch shown; \(y=\mathrm{f}(x)\): intercepts \((-0.2,0)\), \((0,\ln2)\); \(y=\mathrm{f}^{-1}(x)\): intercepts \((\ln2,0)\), \((0,-0.2)\) (b) \(x=64\)

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