2023 MI P1 Q6
2023 MI P1 Q6
8 marks
Prelims
The function \(\mathrm{f}\) is defined by
\(\mathrm{f}:x\mapsto -\frac{x}{2x+1},x\in \mathbb{R},x\ne -\frac{1}{2}\).
- Find \({{\mathrm{f}}^{-1}}\left( x \right)\) and state its domain.[3]
- Hence, or otherwise, find \({{\mathrm{f}}^{2}}\left( x \right)\).[1]
- Deduce the exact value of \({{\mathrm{f}}^{2023}}\left( 5 \right)\).[2]
The function \(\mathrm{g}\) is defined by
\(\mathrm{g}:x\mapsto \frac{1}{ax},x\in \mathbb{R},x>a\),
where \(a\) is a positive constant.
- Determine whether the composite function \(\mathrm{fg}\) exists, justifying your answer.[2]
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Answer:(i) \({\mathrm{f}^{-1}(x) = -\frac{x}{2x+1}\;; \mathbb{R} \setminus \left\{-\frac{1}{2}\right\}}\) (ii) \({\mathrm{f}^2(x) = x}\) (iii) \({-\frac{5}{11}}\) (iv) \({\mathrm{fg} \text{ exists}}\)
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