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If $f(x)=\frac{2 x+3}{3 x-2}, x \neq \frac{2}{3}$ then f of is
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The correct answer is:
an odd function
We have, $\mathrm{f}(\mathrm{x})=\frac{2 \mathrm{x}+3}{3 \mathrm{x}-2}$
$\therefore(\mathrm{fof})(\mathrm{x})=\mathrm{f}[\mathrm{f}(\mathrm{x})]=\mathrm{f}\left(\frac{2 \mathrm{x}+3}{3 \mathrm{x}-2}\right)$
$=\frac{2\left(\frac{2 \mathrm{x}+3}{3 \mathrm{x}-2}\right)+3}{3\left(\frac{2 \mathrm{x}+3}{3 \mathrm{x}-2}\right)-2}=\frac{2(2 \mathrm{x}+3)+3(3 \mathrm{x}-2)}{3(2 \mathrm{x}+3)-2(3 \mathrm{x}-2)}$
$=\frac{4 \mathrm{x}+6+9 \mathrm{x}-6}{6 \mathrm{x}+9-6 \mathrm{x}+4}=\frac{13 \mathrm{x}}{13}=\mathrm{x} \quad \ldots$ is an odd function
$\therefore(\mathrm{fof})(\mathrm{x})=\mathrm{f}[\mathrm{f}(\mathrm{x})]=\mathrm{f}\left(\frac{2 \mathrm{x}+3}{3 \mathrm{x}-2}\right)$
$=\frac{2\left(\frac{2 \mathrm{x}+3}{3 \mathrm{x}-2}\right)+3}{3\left(\frac{2 \mathrm{x}+3}{3 \mathrm{x}-2}\right)-2}=\frac{2(2 \mathrm{x}+3)+3(3 \mathrm{x}-2)}{3(2 \mathrm{x}+3)-2(3 \mathrm{x}-2)}$
$=\frac{4 \mathrm{x}+6+9 \mathrm{x}-6}{6 \mathrm{x}+9-6 \mathrm{x}+4}=\frac{13 \mathrm{x}}{13}=\mathrm{x} \quad \ldots$ is an odd function
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