2025 CJC Promo Q10

2025 CJC Promo Q10

Junior College 1
10 marks

The functions f and g are defined by

where \(\lambda\) is a positive constant.

  1. Show that f has an inverse and find an expression for \(\mathrm{f}^{-1}(x)\).[4]
  2. Find the range of values of \(x\) such that \(\mathrm{f}^{-1}\mathrm{f}(x)=\mathrm{ff}^{-1}(x)\).[1]
    1. Explain why fg does not exist.[1]
    2. By restricting the domain of g to \((0,\sqrt{\lambda})\), find the rule of fg and its range.[4]

Solution:

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Answer:(a) \(\mathrm{f}^{-1}(x)=\dfrac{2x-2\lambda}{x+2}\), \(x>0\) (b) \(0<x<2\) (c)(ii) \(\mathrm{fg}(x)=\dfrac{2x^2}{\lambda+2-x^2}\), range \((0,\lambda)\)

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