The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \frac{1}{x-2}$

Question

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \frac{1}{x-2}$, $x\ge 3$, $\mathrm{g}:x\mapsto \ln x$, $x>0$. Show that the composite function $\mathrm{gf}$ exists.[2] Find $\mathrm{gf}$ in similar form and state its range.[3] Without finding the rule of ${{\left( \mathrm{gf} \right)}^{-1}}$, determine the value of $a$ given that ${{\left( \mathrm{gf} \right)}^{-1}}\left( a \right)=4$.[2]

Accepted Answer

(i) Since ${{	ext{R}}_{	ext{f}}}\subseteq {{	ext{D}}_{	ext{g}}}$, $	ext{gf}$ exists. (ii) $	ext{gf}:x\mapsto \ln \left( \frac{1}{x-2} ight)$, $x\ge 3$ (iii) $a=\ln \left( \frac{1}{2} ight)$ or $-\ln 2$

2024 MI PRU2 P2 Q4

2024 MI PRU2 P2 Q4

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