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Question Number 169922 by cortano1 last updated on 12/May/22

$$Letf\left(x\right)=\frac{2x-7}{x+1}.Compute{f}^{1989}\left(x\right).$$$$note{f}^2\left(x\right)=f\left(f\left(x\right)\right)$$

Answered by floor(10²Eta[1]) last updated on 12/May/22

$$\mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right)=\mathrm{f}\left(\frac{2\mathrm{x}-7}{\mathrm{x}+1}\right)=\frac{2\frac{2\mathrm{x}-7}{\mathrm{x}+1}-7}{\frac{2\mathrm{x}-7}{\mathrm{x}+1}+1}=\frac{-3\mathrm{x}-21}{3\mathrm{x}-6}=\frac{-\mathrm{x}-7}{\mathrm{x}-2}$$$${\mathrm{f}}^3\left(\mathrm{x}\right)=\mathrm{f}\left(\frac{-\mathrm{x}-7}{\mathrm{x}-2}\right)=\frac{-2\frac{\mathrm{x}+7}{\mathrm{x}-2}-7}{\frac{-\mathrm{x}-7}{\mathrm{x}-2}+1}=\frac{9\mathrm{x}}{-9}=-\mathrm{x}$$$$\Rightarrow {\mathrm{f}}^6\left(\mathrm{x}\right)={\mathrm{f}}^3(-\mathrm{x})=\mathrm{x}$$$$\Rightarrow {\mathrm{f}}^{\mathrm{n}}\left(\mathrm{x}\right)=-\mathrm{x},\mathrm{n}=3\mathrm{a},\mathrm{a}\mathrm{is}\mathrm{odd}$$$$\Rightarrow {\mathrm{f}}^{\mathrm{n}}\left(\mathrm{x}\right)=\mathrm{x},\mathrm{n}=3\mathrm{b},\mathrm{b}\mathrm{is}\mathrm{even}$$$$\Rightarrow {\mathrm{f}}^{1989}\left(\mathrm{x}\right)=-\mathrm{x}$$

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