2023 IGCSE Additional Mathematics May/June 0606/23 Q8
2023 IGCSE Additional Mathematics May/June 0606/23 Q8
14 marks
- The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by\(\mathrm{f}(x)=\sec x\qquad\text{for}\hspace{0.5em}\dfrac{\pi}{2}<x<\dfrac{3\pi}{2}\)\(\mathrm{g}(x)=3\left(x^2-1\right)\qquad\text{for all real}\hspace{0.5em}x.\)
- Find the range of \(\mathrm{f}\).[1]
- Solve the equation \(\mathrm{f}^{-1}(x)=\dfrac{2\pi}{3}\).[3]
- Given that \(\mathrm{gf}\) exists, state the domain of \(\mathrm{gf}\).[1]
- Solve the equation \(\mathrm{gf}(x)=1\).[5]
- The function \(\mathrm{h}\) is defined by \(\mathrm{h}(x)=\ln(4-x)\) for \(x<4\). Sketch the graph of \(y=\mathrm{h}(x)\) and hence sketch the graph of \(y=\mathrm{h}^{-1}(x)\). Show the position of any asymptotes and any points of intersection with the coordinate axes.[4]
Solution:
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Answer:(a)(i) \(\mathrm{f}(x)\le-1\) (ii) \(x=-2\) (iii) \(\dfrac{\pi}{2}<x<\dfrac{3\pi}{2}\) (iv) \(x=\dfrac{5\pi}{6}\) or \(x=\dfrac{7\pi}{6}\) (b) Sketch shown; \(\mathrm{h}\): intercepts \((0,\ln4)\) and \((3,0)\), asymptote \(x=4\); \(\mathrm{h}^{-1}\): intercepts \((0,3)\) and \((\ln4,0)\), asymptote \(y=4\)
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