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

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

Commented by jagoll last updated on 20/Mar/20

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

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