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

What is f(x) if : 2f(x)−x f(((2x+3)/(x−2)))=3 when x≠ 2 ?

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

Answered by EDWIN88 last updated on 07/Feb/21

(1) 2f(x)−x f(((2x+3)/(x−2)))= 3 replacing x by ((2x+3)/(x−2)) (2) 2f(((2x+3)/(x−2)))−(((2x+3)/(x−2)))f(x)= 3 then multiply by (2)eq (1) and by x eq(2) ⇒4f(x)−2x f(((2x+3)/(x−2)))= 6 ⇒ 2x f(((2x+3)/(x−2)))−(((2x^2 +3x)/(x−2)))f(x)= 3x ^(−−−−−−−−−−−−−−−−−−−−−−−−) + ⇒ (((4x−8−2x^2 −3x)/(x−2)))f(x)= 6+3x f(x)= (((x−2)(3x+6))/(x−2x^2 −8)) = ((3(x^2 −4))/(−2x^2 +x−8)) f(x)= ((3(4−x^2 ))/(2x^2 −x+8))

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

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