2021 TMJC P1 Q5
2021 TMJC P1 Q5
9 marks
Prelims
It is given that
\(\mathrm{f}\left( x \right)=\left\{ \begin{matrix} & x,\,\,\mathrm{for}-2<;x\le 3, \\ & \sqrt{\left( {{x}^{2}}+kx+3 \right)}+3,\,\,\mathrm{for}\,\,3<x\le 5, \\ \end{matrix} \right.\)
where \(k\) is a real constant such that \(\mathrm{f}\) is continuous for \(-2 < x\le 5\).
- Show that \(k=-4\).[1]
- Sketch the graph of \(y=\mathrm{f}\left( x \right)\). Hence, justify that \(\mathrm{f}\) has an inverse.[3]
- Find \({{\mathrm{f}}^{-1}}\left( x \right)\) in similar form. Leave your answer in the exact form.[4]
- Write down the solution set for \({{\mathrm{f}}^{-1}}\left( x \right)=\mathrm{f}\left( x \right)\).[1]
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Answer:(iii) Piecewise: \(x\) for \(-2<x\leq 3\); \(2+\sqrt{(x-3)^{2}+1}\) for \(3<x\leq 3+2\sqrt{2}\). (iv) \(\{x:-2<x\leq 3\}\)
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