2022 ACJC P1 Q6
2022 ACJC P1 Q6
7 marks
Prelims
The function \(\mathrm{h}\) is defined by \(\mathrm{h}:x\mapsto \left| \ln \left( 2x \right) \right|+1\), \(x\in \mathbb{R}\), \(0 < x\le \lambda \).
- Find the maximum value of \(\lambda \) for which the inverse function \(\mathrm{h}\) exist.[1]
- Using \(\lambda =\frac{1}{2}\),
- Find \({{\mathrm{h}}^{-1}}\) and state its domain,[4]
- sketch the graph of \(y=\mathrm{h}{{\mathrm{h}}^{-1}}\left( x \right)\),[1]
- find the solution set for \(\mathrm{h}\left( x \right)={{\mathrm{h}}^{-1}}\left( x \right)\).[1]
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Answer:(i) \(\lambda=\frac12\) (ii)(a) \(h^{-1}(x)=\frac12\mathrm e^{1-x}\), \(\mathrm D_{h^{-1}}=[1,\infty)\) (ii)(b) \(y=x\) for \(x\ge1\) (ii)(c) \(\varnothing\)
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