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2022 IGCSE Additional Mathematics October/November 0606/13 Q6

2022 IGCSE Additional Mathematics October/November 0606/13 Q6

7 marks

A function \(\mathrm{f}(x)\) is such that \(\mathrm{f}(x)=\mathrm{e}^{3x}-4\), for \(x\in\mathbb{R}\).

  1. Find the range of \(\mathrm{f}\).[1]
  2. Find an expression for \(\mathrm{f}^{-1}(x)\).[2]
  3. On the axes, sketch the graphs of \(y=\mathrm{f}(x)\) and \(y=\mathrm{f}^{-1}(x)\) stating the exact values of the intercepts with the coordinate axes.[4]

Solution:

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Answer:(a) \(\mathrm{f}(x)>-4\) (b) \(\mathrm{f}^{-1}(x)=\dfrac13\ln(x+4)\) (c) Sketch shown; \(y=\mathrm{f}(x)\) meets the axes at \((0,-3)\) and \(\left(\dfrac13\ln4,0\right)\) with asymptote \(y=-4\); \(y=\mathrm{f}^{-1}(x)\) meets them at \((-3,0)\) and \(\left(0,\dfrac13\ln4\right)\) with asymptote \(x=-4\)

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