2024 IGCSE Additional Mathematics October/November 0606/12 Q4

2024 IGCSE Additional Mathematics October/November 0606/12 Q4

13 marks

The function \(\mathrm{f}\) is such that \(\mathrm{f}(x)=4\ln(3x-2)\), for \(x>a\), where \(a\) is as small as possible.

    1. Write down the value of \(a\).[1]
    2. Write down the range of \(\mathrm{f}\).[1]
    3. Find \(\mathrm{f}^{-1}(x)\), stating its domain and range.[4]
    4. On the axes sketch the graphs of \(y=\mathrm{f}(x)\) and \(y=\mathrm{f}^{-1}(x)\), stating the intercepts with the axes.
  2. Given that \(\mathrm{g}(x)=(2x+1)^{\frac12}+4\), for \(x>0\), solve the equation \(\mathrm{gg}(x)=9\).[3]

Solution:

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Answer:(a)(i) \(\dfrac23\). (ii) \(\mathbb R\). (iii) \(\mathrm{f}^{-1}(x)=\dfrac{\mathrm{e}^{\frac x4}+2}3\), domain \(\mathbb R\), range \((\dfrac23,\infty)\). (iv) Sketch shown; intercepts \((1,0)\), \((0,1)\). (b) \(31.5\).

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