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2021 IGCSE Additional Mathematics October/November 0606/21 Q9

2021 IGCSE Additional Mathematics October/November 0606/21 Q9

9 marks

The following functions are defined for \(x>1\).

\(\mathrm{f}(x)=\dfrac{x+3}{x-1}\qquad\mathrm{g}(x)=1+x^2\)

  1. Find \(\mathrm{fg}(x)\).[2]
  2. Find \(\mathrm{g}^{-1}(x)\).[2]
  3. DO NOT USE A CALCULATOR IN THIS PART OF THE QUESTION.
    Solve the equation \(\mathrm{f}(x)=\mathrm{g}(x)\).[5]

Solution:

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Answer:(a) \(\mathrm{fg}(x)=\dfrac{x^2+4}{x^2}\) (b) \(\mathrm{g}^{-1}(x)=\sqrt{x-1}\) (c) \(x=2\) only, since \(x^3-x^2-4=(x-2)\left(x^2+x+2\right)\) and \(x^2+x+2=0\) has no real roots

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