2023 IGCSE Additional Mathematics February/March 0606/22 Q8
2023 IGCSE Additional Mathematics February/March 0606/22 Q8
13 marks
The function \(\mathrm{f}\) is defined for \(x\ge0\) by \(\mathrm{f}(x)=5-2\mathrm{e}^{-x}\).
- Find the domain of \(\mathrm{f}^{-1}\).[2]
- Solve \(\mathrm{ff}^{-1}(x)=\sqrt{5x-4}\).[3]
- On the axes, sketch the graph of \(y=\mathrm{f}(x)\) and hence sketch the graph of \(y=\mathrm{f}^{-1}(x)\). Show clearly the positions of any points where your graphs meet the coordinate axes and the positions of any asymptotes.[4]
- The function \(\mathrm{g}\) is defined for \(0\le x\le0.2\) by \(\mathrm{g}(x)=\dfrac{3}{1-x}\).Find and simplify an expression for \(\mathrm{f}^{-1}\mathrm{g}(x)\).[4]
Solution:
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Answer:(a)(i) \(3\le x<5\) (ii) \(x=4\) (iii) Sketch shown; \(\mathrm{f}\): intercept \((0,3)\), asymptote \(y=5\); \(\mathrm{f}^{-1}\): intercept \((3,0)\), asymptote \(x=5\) (b) \(\mathrm{f}^{-1}\mathrm{g}(x)=\ln\left(\dfrac{2-2x}{2-5x}\right)\)
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