2020 IGCSE Additional Mathematics May/June 0606/21 Q11
2020 IGCSE Additional Mathematics May/June 0606/21 Q11
10 marks
The function \(\mathrm{f}\) is defined by \(\mathrm{f}(x)=\ln(2x+1)\) for \(x\ge0\).
- Sketch the graph of \(y=\mathrm{f}(x)\) and hence sketch the graph of \(y=\mathrm{f}^{-1}(x)\) on the axes below.[3]
The function \(\mathrm{g}\) is defined by \(\mathrm{g}(x)=(x-4)^2+1\) for \(x\le4\).
- Find an expression for \(\mathrm{g}^{-1}(x)\) and state its domain and range.[4]
- Find and simplify an expression for \(\mathrm{fg}(x)\).[2]
- Explain why the function \(\mathrm{gf}\) does not exist.[1]
Solution:
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Answer:(a) Sketch shown: both graphs start at \(O\) and are reflections in \(y=x\); \(\mathrm{f}^{-1}(x)=\dfrac{\mathrm{e}^x-1}{2}\) (b)(i) \(\mathrm{g}^{-1}(x)=4-\sqrt{x-1}\), domain \(x\ge1\), range \(\mathrm{g}^{-1}(x)\le4\) (ii) \(\mathrm{fg}(x)=\ln\left(2x^2-16x+35\right)\) (iii) The range of \(\mathrm{f}\), \(\mathrm{f}(x)\ge0\), includes values outside the domain \(x\le4\) of \(\mathrm{g}\).
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