2022 IGCSE Additional Mathematics October/November 0606/12 Q8
2022 IGCSE Additional Mathematics October/November 0606/12 Q8
10 marks
A function \(\mathrm{f}(x)\) is such that \(\mathrm{f}(x)=\ln(2x+3)+\ln4\), for \(x>a\), where \(a\) is a constant.
- Write down the least possible value of \(a\).[1]
- Using your value of \(a\), write down the range of \(\mathrm{f}\).[1]
- Using your value of \(a\), find \(\mathrm{f}^{-1}(x)\), stating its range.[4]
- On the axes below, sketch the graphs of \(y=\mathrm{f}(x)\) and \(y=\mathrm{f}^{-1}(x)\), stating the exact intercepts of each graph with the coordinate axes. Label each of your graphs.[4]
Solution:
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Answer:(a) \(a=-1.5\) (b) \(\mathrm{f}(x)\in\mathbb{R}\) (c) \(\mathrm{f}^{-1}(x)=\dfrac{\mathrm{e}^x-12}{8}\), range \(\mathrm{f}^{-1}(x)>-1.5\) (d) Sketch shown; \(y=\mathrm{f}(x)\): intercepts \(\left(-\dfrac{11}{8},0\right)\), \((0,\ln12)\); \(y=\mathrm{f}^{-1}(x)\): intercepts \((\ln12,0)\), \(\left(0,-\dfrac{11}{8}\right)\)
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