2023 IGCSE Additional Mathematics May/June 0606/21 Q8
2023 IGCSE Additional Mathematics May/June 0606/21 Q8
11 marks
- The diagram shows the graph of \(y=\mathrm{f}(x)\) where \(\mathrm{f}\) is defined by \(\mathrm{f}(x)=\dfrac{3x}{\sqrt{5x+1}}\) for \(0\le x\le3\).
- Given that \(\mathrm{f}\) is a one-one function, find the domain and range of \(\mathrm{f}^{-1}\).[3]
- Solve the equation \(\mathrm{f}(x)=x\).[2]
- On the diagram above, sketch the graph of \(y=\mathrm{f}^{-1}(x)\).[2]
- The functions \(\mathrm{g}\) and \(\mathrm{h}\) are defined by\(\mathrm{g}(x)=\sqrt[3]{8x^3+3}\quad\text{for}\hspace{0.5em}x\ge1,\)\(\mathrm{h}(x)=\mathrm{e}^{4x}\quad\text{for}\hspace{0.5em}x\ge k.\)
- Find an expression for \(\mathrm{g}^{-1}(x)\).[2]
- State the least value of the constant \(k\) such that \(\mathrm{gh}(x)\) can be formed.[1]
- Find and simplify an expression for \(\mathrm{gh}(x)\).[1]
Solution:
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Answer:(a)(i) Domain \(0\le x\le\dfrac94\), range \(0\le\mathrm{f}^{-1}(x)\le3\) (ii) \(x=0\) or \(x=\dfrac85\) (iii) Sketch shown: reflection of \(y=\mathrm{f}(x)\) in \(y=x\), from \((0,0)\) to \(\left(\dfrac94,3\right)\) (b)(i) \(\mathrm{g}^{-1}(x)=\dfrac12\sqrt[3]{x^3-3}\) (ii) \(k=0\) (iii) \(\mathrm{gh}(x)=\sqrt[3]{8\mathrm{e}^{12x}+3}\)
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