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Question Number 115997 by bemath last updated on 30/Sep/20

Answered by bobhans last updated on 30/Sep/20

$$\left(1\right)f\left(\frac{1}{x}\right)+f(1-x)=x\to f(1-x)=x-f\left(\frac{1}{x}\right)$$$$replace1-xbyx$$$$\left(2\right)f\left(\frac{1}{1-x}\right)+f\left(x\right)=1-x$$$$replace\frac{1}{x}by1-x$$$$\left(3\right)f(1-x)+f(1-\frac{1}{1-x})=1-\frac{1}{1-x}$$$$f(1-x)+f\left(\frac{x}{x-1}\right)=\frac{x}{x-1}$$$$\iff -f\left(\frac{1}{x}\right)+f\left(\frac{x}{x-1}\right)=\frac{x}{x-1}-\frac{x^2-x}{x-1}$$$$\iff f\left(\frac{x}{x-1}\right)-f\left(\frac{1}{x}\right)=\frac{-x^2+2x}{x-1}$$$$continue$$

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