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2026 RVHS P1 Q9

2026 RVHS P1 Q9

Junior College 2
9 marks
  1. The function \(\mathrm{f}\) is defined by \(\mathrm{f}:x\mapsto4+e^{-x}\), where \(0<x\leq4\).
    1. Show that \(\mathrm{f}^{-1}\) exists.[1]
    2. Find the \(\mathrm{f}^{-1}(x)\) and the domain of \(\mathrm{f}^{-1}\).[3]
  2. The function \(\mathrm{g}\) is defined by
    \[\mathrm{g}:x\mapsto\mathrm{f}(x),\text{ for}\hspace{0.5em}0<x\leq4,\]
    and that \(\mathrm{g}(x)=\mathrm{g}(x+4)\) for all real values of \(x\).
    1. Explain why the composite function \(\mathrm{g}^2\) exists.[1]
    2. Find the exact value of \(\mathrm{g}^2(2)\).[2]
    3. State the range of the composite function \(\mathrm{g}^2\).[2]

Solution:

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Answer:(a)(ii) \(\mathrm{f}^{-1}(x)=-\ln(x-4)\), domain \([4+e^{-4},5)\) (b)(ii) \(4+e^{-e^{-2}}\) (iii) \(\left(4+e^{-1},4+e^{-e^{-4}}\right]\)

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