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Question Number 85233 by jagoll last updated on 20/Mar/20

what is domain of x for f(x)+2f(1−x)= x^2

$$\mathrm{what}\:\mathrm{is}\:\mathrm{domain}\:\:\mathrm{of}\:\mathrm{x}\:\mathrm{for} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)+\mathrm{2f}\left(\mathrm{1}−\mathrm{x}\right)=\:\mathrm{x}^{\mathrm{2}} \\ $$

Commented by jagoll last updated on 20/Mar/20

replace 1−x by x f(1−x)+2f(x) = (1−x)^2 (ii) multiply by 2 eq (ii) 4f(x)+2f(1−x)= 2x^2 −4x+2 f(x)+2f(1−x) = x^2 (ii)−(i) 3f(x) = x^2 −4x+2 f(x) = ((x^2 −4x+2)/3) , ∀x∈R

$$\mathrm{replace}\:\mathrm{1}−\mathrm{x}\:\mathrm{by}\:\mathrm{x}\: \\ $$$$\mathrm{f}\left(\mathrm{1}−\mathrm{x}\right)+\mathrm{2f}\left(\mathrm{x}\right)\:=\:\left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{2}} \:\left(\mathrm{ii}\right) \\ $$$$\mathrm{multiply}\:\mathrm{by}\:\mathrm{2}\:\mathrm{eq}\:\left(\mathrm{ii}\right) \\ $$$$\mathrm{4f}\left(\mathrm{x}\right)+\mathrm{2f}\left(\mathrm{1}−\mathrm{x}\right)=\:\mathrm{2x}^{\mathrm{2}} −\mathrm{4x}+\mathrm{2} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)+\mathrm{2f}\left(\mathrm{1}−\mathrm{x}\right)\:=\:\mathrm{x}^{\mathrm{2}} \\ $$$$\left(\mathrm{ii}\right)−\left(\mathrm{i}\right)\: \\ $$$$\mathrm{3f}\left(\mathrm{x}\right)\:=\:\mathrm{x}^{\mathrm{2}} −\mathrm{4x}+\mathrm{2} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{4x}+\mathrm{2}}{\mathrm{3}}\:,\:\forall\mathrm{x}\in\mathbb{R} \\ $$

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