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Question Number 120819 by bramlexs22 last updated on 03/Nov/20

$$\mathrm{If}\mathrm{f}\left(\mathrm{x}\right)+2\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)+3\mathrm{f}\left(\frac{\mathrm{x}}{\mathrm{x}-1}\right)=\mathrm{x}$$$$\mathrm{what}\mathrm{is}\mathrm{f}\left(\mathrm{x}\right)?$$

Commented by liberty last updated on 03/Nov/20

$$\left(\mathrm{i}\right)\mathrm{f}\left(\mathrm{x}\right)+2\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)+3\mathrm{f}\left(\frac{\mathrm{x}}{\mathrm{x}-1}\right)=\mathrm{x}$$$$\left(\mathrm{ii}\right)\mathrm{replace}\mathrm{x}\mathrm{with}\frac{1}{\mathrm{x}},\mathrm{give}$$$$\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)+2\mathrm{f}\left(\mathrm{x}\right)+3\mathrm{f}\left(\frac{1}{1-\mathrm{x}}\right)=\frac{1}{\mathrm{x}}$$$$\left(\mathrm{iii}\right)\mathrm{replace}\mathrm{x}\mathrm{with}1-\mathrm{x},\mathrm{give}$$$$\mathrm{f}(1-\mathrm{x})+2\mathrm{f}\left(\frac{1}{1-\mathrm{x}}\right)+3\mathrm{f}\left(\frac{\mathrm{x}-1}{\mathrm{x}}\right)=1-\mathrm{x}$$$$\left(\mathrm{iv}\right)\mathrm{replace}\mathrm{x}\mathrm{with}\frac{1}{\mathrm{x}-1},\mathrm{give}$$$$\mathrm{f}\left(\frac{1}{1-\mathrm{x}}\right)+2\mathrm{f}(1-\mathrm{x})+3\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)=\frac{1}{1-\mathrm{x}}$$$$\left(\mathrm{v}\right)\mathrm{replace}\mathrm{x}\mathrm{with}\frac{\mathrm{x}}{\mathrm{x}-1},\mathrm{give}$$$$\mathrm{f}\left(\frac{\mathrm{x}}{\mathrm{x}-1}\right)+2\mathrm{f}\left(\frac{\mathrm{x}-1}{\mathrm{x}}\right)+3\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{x}}{\mathrm{x}-1}$$$$\left(\mathrm{vi}\right)\mathrm{replace}\mathrm{x}\mathrm{with}\frac{\mathrm{x}-1}{\mathrm{x}}\mathrm{give}$$$$\mathrm{f}\left(\frac{\mathrm{x}-1}{\mathrm{x}}\right)+2\mathrm{f}\left(\frac{\mathrm{x}}{\mathrm{x}-1}\right)+3\mathrm{f}(1-\mathrm{x})=\frac{\mathrm{x}-1}{\mathrm{x}}$$$$\mathrm{adding}\mathrm{all}\mathrm{equations}\mathrm{and}\mathrm{solving}\mathrm{for}$$$$\mathrm{f}\left(\mathrm{x}\right)\mathrm{from}\mathrm{this}\mathrm{system}\mathrm{give}$$$$\because \mathrm{f}\left(\mathrm{x}\right)=\frac{{2\mathrm{x}}^3+{\mathrm{x}}^2+5\mathrm{x}-2}{24({\mathrm{x}}^2-\mathrm{x})}$$

Commented by bramlexs22 last updated on 03/Nov/20

$$\mathrm{great}...$$

Commented by Ar Brandon last updated on 03/Nov/20

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