2025 SAJC Promo Q5

2025 SAJC Promo Q5

Junior College 1
9 marks
  1. The function \(\mathrm{f}\) is defined as follows.

    \(\mathrm{f}:x\mapsto a(x-1)^2+b,\qquad x\in\mathbb{R},\ x>0.\)

    1. Given that \(a>0\) and \(0<b<1\), show that \(\mathrm{f}^2\) exists.[2]
    2. Hence determine the range of \(\mathrm{f}^2\), leaving your answer in terms of \(b\).[1]
  2. The function \(\mathrm{g}\) is defined as follows.

    \(\mathrm{g}:x\mapsto \frac{x+4}{2x-1},\qquad x\in\mathbb{R},\ x \ne \frac{1}{2}.\)

    1. Find \(\mathrm{g}^{-1}(x)\) and state the domain of \(\mathrm{g}^{-1}\).[3]
    2. Find the values of \(c\) such that \(\mathrm{g}^{2024}(c)=\mathrm{g}^{-1}(c)\). Explain your answers clearly.[3]

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Answer:\(R_{\mathrm{f}^2}=[b,\infty)\); \(\mathrm{g}^{-1}(x)=\dfrac{x+4}{2x-1}\); \(c=-1\) or \(2\)

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