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2022 IGCSE Additional Mathematics October/November 0606/21 Q9

2022 IGCSE Additional Mathematics October/November 0606/21 Q9

10 marks

The functions \(\mathrm{f}(x)\) and \(\mathrm{g}(x)\) are defined as follows for \(x>-\dfrac13\) by

\(\mathrm{f}(x)=x^2+1,\)

\(\mathrm{g}(x)=\ln(3x+2).\)

  1. Find \(\mathrm{fg}(x)\).[1]
  2. Solve the equation \(\mathrm{fg}(x)=5\) giving your answer in exact form.[3]
  3. Solve the equation \(\mathrm{gg}(x)=1\).[6]

Solution:

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Answer:(a) \(\mathrm{fg}(x)=\left(\ln(3x+2)\right)^2+1\) (b) \(x=\dfrac{\mathrm{e}^2-2}{3}\) only (c) \(x=\dfrac{\mathrm{e}^{\frac{\mathrm{e}-2}{3}}-2}{3}=-0.243\) (3 s.f.)

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