The function $\mathrm f$ is given by \[\mathrm f:x\mapsto\ln x,\quad x\in\mathbb R,\quad0

Question

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]

Accepted Answer

(a) $y=\ln x,02$ also violates $D_{\mathrm f}$.

2026 CJC Promo Q11

2026 CJC Promo Q11

Solution:

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