2026 CJC Promo Q11
2026 CJC Promo Q11
Junior College 1
12 marks
Promo
The function \(\mathrm f\) is given by
\[\mathrm f:x\mapsto\ln x,\quad x\in\mathbb R,\quad0<x\le2.\]
- Sketch the curve of \(y=\mathrm f(x)\).[2]
- Show that \(\mathrm f^{-1}\) exists.[1]
- Find \(\mathrm f^{-1}(x)\) and state its domain.[3]
The function \(\mathrm g\) is given by
\[\mathrm g:x\mapsto\frac12\cos x+1,\quad x\in\mathbb R,\quad0\le x\le2\pi.\]
- Explain why the composite function \(\mathrm{fg}\) exists.[1]
- Find the value of \(\mathrm{fg}\left(\dfrac{3\pi}2\right)\).[2]
- Find the exact value of \(b\) such that \(\mathrm g^{-1}\mathrm f(b)=\dfrac\pi3\).[3]
Solution:
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Answer:(a) \(y=\ln x,0<x\le2\). (b) One-to-one. (c) \(\mathrm f^{-1}(x)=e^x,\ x\le\ln2\). (d) \(R_{\mathrm g}\subseteq D_{\mathrm f}\). (e) \(0\). (f) Invalid as printed: \(\mathrm g^{-1}\) does not exist; the supplied \(b=e^{5/4}>2\) also violates \(D_{\mathrm f}\).
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