2024 IGCSE Additional Mathematics May/June 0606/22 Q10

2024 IGCSE Additional Mathematics May/June 0606/22 Q10

8 marks

The functions f and fg are defined by

\(\mathrm{f}(x)=\mathrm{e}^{x^2+3}\quad\text{for}\hspace{0.5em}x<0\)

\(\mathrm{fg}(x)=\mathrm{e}^{2x}\quad\text{for}\hspace{0.5em}x>\dfrac32.\)

  1. Explain why \(\mathrm{f}^{-1}\) exists.[1]
  2. Find an expression for \(\mathrm{f}^{-1}(x)\) and state the domain and range of \(\mathrm{f}^{-1}\).[5]
  3. Hence find and simplify an expression for \(\mathrm{g}(x)\).[2]

Solution:

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Answer:(a) \(\mathrm{f}\) is one-to-one. (b) \(\mathrm{f}^{-1}(x)=-\sqrt{\ln x-3}\); domain \(x>\mathrm{e}^3\), range \(<0\). (c) \(\mathrm{g}(x)=-\sqrt{2x-3}\).

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