The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows: \(\mathrm{f}:x\mapsto \frac{

Question

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows: $\mathrm{f}:x\mapsto \frac{1}{1-x}$, $x\in \mathbb{R}$, $x\ne 0$ or $1$, $\mathrm{g}:x\mapsto 1-\frac{1}{x}$, $x\in \mathbb{R}$, $x\ne 0$ or $1$. It is given that the range of $\mathrm{g}$ is $\mathbb{R}\backslash \left\{ 0,1 \right\}$. Prove that the composite function $\mathrm{fg}$ exists.[2] Express $\mathrm{fg}$ in a similar form and hence describe the relationship between $\mathrm{f}$ and $\mathrm{g}$.[3] Evaluate ${{\mathrm{f}}^{2011}}{{\mathrm{g}}^{1994}}\left( \frac{1}{2} \right)$.[3]

Accepted Answer

(ii) $\mathrm{fg}:x\mapsto x$, $x\in \mathbb{R}$, $x\ne 0$ or $1$, $\mathrm{f}$ and $\mathrm{g}$ are inverse functions (iii) $-1$

Finding \({\mathrm{f}^n}\) where \({n}\) is large Q1

Finding \({\mathrm{f}^n}\) where \({n}\) is large Q1

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