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Question Number 96829 by bobhans last updated on 05/Jun/20

If 2f(x) + f(1−x) = x^2 . determine f(x)

$$\mathrm{If}\:\mathrm{2f}\left(\mathrm{x}\right)\:+\:\mathrm{f}\left(\mathrm{1}−\mathrm{x}\right)\:=\:\mathrm{x}^{\mathrm{2}} .\:\mathrm{determine}\:\mathrm{f}\left(\mathrm{x}\right) \\ $$

Commented by john santu last updated on 05/Jun/20

replace x by 1−x into eq(1) (2) 2f(1−x) +f(x) = (1−x)^2 2f(1−x) +f(x) = x^2 −2x+1 multiply eq (1) by 2 (1) 2f(1−x)+4f(x)=2x^2 subtract eq(2) by eq(1) 3f(x) = 2x^2 −(x^2 −2x+1) 3f(x) = x^2 +2x−1 ∴ f(x) = ((x^2 +2x−1)/3) .

$$\mathrm{replace}\:\mathrm{x}\:\mathrm{by}\:\mathrm{1}−\mathrm{x}\:\mathrm{into}\:\mathrm{eq}\left(\mathrm{1}\right) \\ $$$$\left(\mathrm{2}\right)\:\mathrm{2f}\left(\mathrm{1}−\mathrm{x}\right)\:+\mathrm{f}\left(\mathrm{x}\right)\:=\:\left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{2}} \\ $$$$\mathrm{2f}\left(\mathrm{1}−\mathrm{x}\right)\:+\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{x}^{\mathrm{2}} −\mathrm{2x}+\mathrm{1} \\ $$$$\mathrm{multiply}\:\mathrm{eq}\:\left(\mathrm{1}\right)\:\mathrm{by}\:\mathrm{2}\: \\ $$$$\left(\mathrm{1}\right)\:\mathrm{2f}\left(\mathrm{1}−\mathrm{x}\right)+\mathrm{4f}\left(\mathrm{x}\right)=\mathrm{2x}^{\mathrm{2}} \\ $$$$\mathrm{subtract}\:\mathrm{eq}\left(\mathrm{2}\right)\:\mathrm{by}\:\mathrm{eq}\left(\mathrm{1}\right) \\ $$$$\mathrm{3f}\left(\mathrm{x}\right)\:=\:\mathrm{2x}^{\mathrm{2}} −\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2x}+\mathrm{1}\right) \\ $$$$\mathrm{3f}\left(\mathrm{x}\right)\:=\:\mathrm{x}^{\mathrm{2}} +\mathrm{2x}−\mathrm{1} \\ $$$$\therefore\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}−\mathrm{1}}{\mathrm{3}}\:. \\ $$

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