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Question Number 131641 by liberty last updated on 07/Feb/21

$$\mathrm{What}\mathrm{is}\mathrm{f}\left(\mathrm{x}\right)\mathrm{if}:2\mathrm{f}\left(\mathrm{x}\right)-\mathrm{x}\mathrm{f}\left(\frac{2\mathrm{x}+3}{\mathrm{x}-2}\right)=3$$$$\mathrm{when}\mathrm{x}\ne 2?$$

Answered by EDWIN88 last updated on 07/Feb/21

$$\left(1\right)2\mathrm{f}\left(\mathrm{x}\right)-\mathrm{x}\mathrm{f}\left(\frac{2\mathrm{x}+3}{\mathrm{x}-2}\right)=3$$$$\mathrm{replacing}\mathrm{x}\mathrm{by}\frac{2\mathrm{x}+3}{\mathrm{x}-2}$$$$\left(2\right)2\mathrm{f}\left(\frac{2\mathrm{x}+3}{\mathrm{x}-2}\right)-\left(\frac{2\mathrm{x}+3}{\mathrm{x}-2}\right)\mathrm{f}\left(\mathrm{x}\right)=3$$$$\mathrm{then}\mathrm{multiply}\mathrm{by}\left(2\right)\mathrm{eq}\left(1\right)\mathrm{and}\mathrm{by}\mathrm{x}\mathrm{eq}\left(2\right)$$$$\Rightarrow 4\mathrm{f}\left(\mathrm{x}\right)-2\mathrm{x}\mathrm{f}\left(\frac{2\mathrm{x}+3}{\mathrm{x}-2}\right)=6$$$$\Rightarrow 2\mathrm{x}\mathrm{f}\left(\frac{2\mathrm{x}+3}{\mathrm{x}-2}\right)-\left(\frac{{2\mathrm{x}}^2+3\mathrm{x}}{\mathrm{x}-2}\right)\mathrm{f}\left(\mathrm{x}\right)=3\mathrm{x}$$$$\stackrel{------------------------}{}+$$$$\Rightarrow \left(\frac{4\mathrm{x}-8-{2\mathrm{x}}^2-3\mathrm{x}}{\mathrm{x}-2}\right)\mathrm{f}\left(\mathrm{x}\right)=6+3\mathrm{x}$$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{(\mathrm{x}-2)(3\mathrm{x}+6)}{\mathrm{x}-{2\mathrm{x}}^2-8}=\frac{3({\mathrm{x}}^2-4)}{-{2\mathrm{x}}^2+\mathrm{x}-8}$$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{3(4-{\mathrm{x}}^2)}{{2\mathrm{x}}^2-\mathrm{x}+8}$$

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