2022 NJC Promo Q8
2022 NJC Promo Q8
12 marks
Promo
The function \(\mathrm{f}\) is defined by
\({\mathrm{f}(x) = \begin{cases} x-4 & \mathrm{for } \quad\quad\quad x \leq 1 \\ \cos x & \mathrm{for } \quad 1 < x \leq \pi \end{cases}}\)
- Show that \(\mathrm{f}\) has an inverse.[3]
- Find \({{\mathrm{f}}^{-1}}\).[3]
- Evaluate \({{\mathrm{f}}^{2}}\left( \frac{\pi }{2} \right)\) and find \({{\mathrm{f}}^{2}}\).[4]
- Find \({{\mathrm{f}}^{2022}}\left( 0 \right)\).[2]
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Answer:(i) \(\mathrm{f}\) is one-one; \(\mathrm{f}^{-1}\) exists (ii) \(\displaystyle \mathrm{f}^{-1}(x)=\begin{cases}x+4,&x\leq-3,\\ \cos^{-1}x,&-1\leq x<\cos1\end{cases}\) (iii) \(\displaystyle \mathrm{f}^{2}\left(\frac{\pi}{2}\right)=-4\), \(\displaystyle \mathrm{f}^{2}(x)=\begin{cases}x-8,&x\leq1,\\ \cos x-4,&1<x\leq\pi\end{cases}\) (iv) \(-8088\)
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