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Question Number 114586 by Khalmohmmad last updated on 19/Sep/20

f(x)−2f((1/2))=3x−2 f(x)=?

$$\mathrm{f}\left(\mathrm{x}\right)−\mathrm{2f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\mathrm{3x}−\mathrm{2} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=? \\ $$

Commented by Dwaipayan Shikari last updated on 19/Sep/20

f(x)−2f((1/x))=3x−2→(a) f((1/x))−2f(x)=(3/x)−2 2f((1/x))−4f(x)=(6/x)−4→(b) −3f(x)=3x+(6/x)−6 f(x)=2−x+(2/x)

$${f}\left({x}\right)−\mathrm{2}{f}\left(\frac{\mathrm{1}}{{x}}\right)=\mathrm{3}{x}−\mathrm{2}\rightarrow\left({a}\right) \\ $$$${f}\left(\frac{\mathrm{1}}{{x}}\right)−\mathrm{2}{f}\left({x}\right)=\frac{\mathrm{3}}{{x}}−\mathrm{2} \\ $$$$\mathrm{2}{f}\left(\frac{\mathrm{1}}{{x}}\right)−\mathrm{4}{f}\left({x}\right)=\frac{\mathrm{6}}{{x}}−\mathrm{4}\rightarrow\left({b}\right) \\ $$$$−\mathrm{3}{f}\left({x}\right)=\mathrm{3}{x}+\frac{\mathrm{6}}{{x}}−\mathrm{6} \\ $$$${f}\left({x}\right)=\mathrm{2}−{x}+\frac{\mathrm{2}}{{x}} \\ $$

Commented by Aziztisffola last updated on 19/Sep/20

f((1/2)) or f((1/x))?

$$\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:\mathrm{or}\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)? \\ $$

Answered by Olaf last updated on 19/Sep/20

f(x) = 3x−2+2f((1/2)) (1) for x = (1/2) : f((1/2))−2f((1/2)) = 3(1/2)−2 = −(1/2) −f((1/2)) = −(1/2) f((1/2)) = (1/2) (1) : f(x) = 3x−2+2((1/2)) = 3x−1 f(x) = 3x−1

$${f}\left({x}\right)\:=\:\mathrm{3}{x}−\mathrm{2}+\mathrm{2}{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:\left(\mathrm{1}\right) \\ $$$$\mathrm{for}\:{x}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:: \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)−\mathrm{2}{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\:\mathrm{3}\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{2}\:=\:−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$−{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\:−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left(\mathrm{1}\right)\::\:{f}\left({x}\right)\:=\:\mathrm{3}{x}−\mathrm{2}+\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\:\mathrm{3}{x}−\mathrm{1} \\ $$$${f}\left({x}\right)\:=\:\mathrm{3}{x}−\mathrm{1} \\ $$

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