N2019 P1 Q5

N2019 P1 Q5

Junior College 2
6 marks
Free

The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by

\(\mathrm{f}(x)=\mathrm{e}^{2x}-4,\quad x\in\mathbb{R},\)

\(\mathrm{g}(x)=x+2,\quad x\in\mathbb{R}.\)

  1. Find \(\mathrm{f}^{-1}(x)\) and state its domain.[3]
  2. Find the exact solution of \(\mathrm{fg}(x)=5\), giving your answer in its simplest form.[3]

Default solution

  1. \(y=\mathrm{e}^{2x}-4\Rightarrow y+4=\mathrm{e}^{2x}\)
    \(x=\frac12\ln(y+4)\)
    \(\mathrm{f}^{-1}(x)=\frac12\ln(x+4),\quad x>-4\)
  2. \(\mathrm{fg}(x)=\mathrm{e}^{2(x+2)}-4=5\)
    \(2x+4=\ln9=2\ln3\)
    \(\therefore x=\ln3-2\)
Answer:(i) \(\mathrm{f}^{-1}(x)=\frac12\ln(x+4)\), domain \(x>-4\). (ii) \(x=\ln3-2\).

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