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

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

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

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

Commented by Aziztisffola last updated on 19/Sep/20

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

Answered by Olaf last updated on 19/Sep/20

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

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