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Let $f(x)=p x+q$ and $g(x)=m x+n$. Then $f(g(x))=g(f(x))$ is equivalent to
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Verified Answer
The correct answer is:
$f(\mathrm{n})=\mathrm{g}(\mathrm{q})$
$\begin{array}{l}\mathrm{f}(\mathrm{x})=\mathrm{px}+\mathrm{q}, \mathrm{g}(\mathrm{x})=\mathrm{mx}+\mathrm{n} \\ \mathrm{f}(\mathrm{g}(\mathrm{x}))=\mathrm{g}(\mathrm{f}(\mathrm{x})) \\ \Rightarrow \mathrm{f}(\mathrm{mx}+\mathrm{n})=\mathrm{g}(\mathrm{px}+\mathrm{q}) \\ & \Rightarrow \mathrm{p}(\mathrm{mx}+\mathrm{n})+\mathrm{q}=\mathrm{m}(\mathrm{px}+\mathrm{q})+\mathrm{n} \\ & \Rightarrow \mathrm{pmx}+\mathrm{pn}+\mathrm{q}=\mathrm{pmx}+\mathrm{mq}+\mathrm{n} \\ & \Rightarrow \mathrm{pn}+\mathrm{q}=\mathrm{mq}+\mathrm{n} \\ & \Rightarrow \mathrm{f}(\mathrm{n})=\mathrm{g}(\mathrm{q})\end{array}$
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