2023 IGCSE Additional Mathematics May/June 0606/12 Q8
2023 IGCSE Additional Mathematics May/June 0606/12 Q8
10 marks
It is given that \(\mathrm{f}(x)=2\ln(3x-4)\) for \(x>a\).
- Write down the least possible value of \(a\).[1]
- Write down the range of \(\mathrm{f}\).[1]
- It is given that the equation \(\mathrm{f}(x)=\mathrm{f}^{-1}(x)\) has two solutions. (You do not need to solve this equation). Using your answer to part (a), sketch the graphs of \(y=\mathrm{f}(x)\) and \(y=\mathrm{f}^{-1}(x)\) on the axes below, stating the coordinates of the points where the graphs meet the axes.[4]
It is given that \(\mathrm{g}(x)=2x-3\) for \(x\ge3\).
- Find an expression for \(\mathrm{g}(\mathrm{g}(x))\).[1]
- Hence solve the equation \(\mathrm{fg}(\mathrm{g}(x))=4\) giving your answer in exact form.[3]
Solution:
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Answer:(a) \(a=\dfrac43\) (b) \(\mathrm{f}(x)\in\mathbb{R}\) (c) Sketch shown; \(y=\mathrm{f}(x)\) meets the axes at \(\left(\dfrac53,0\right)\), \(y=\mathrm{f}^{-1}(x)\) at \(\left(0,\dfrac53\right)\) (d)(i) \(\mathrm{g}(\mathrm{g}(x))=4x-9\) (ii) \(x=\dfrac{\mathrm{e}^2+31}{12}\)
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