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Question Number 110436 by bemath last updated on 29/Aug/20

$$\mathrm{If}2\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)=3\mathrm{x}$$$$\mathrm{what}\mathrm{is}\mathrm{f}\left(\mathrm{x}\right)$$

Answered by john santu last updated on 29/Aug/20

$$replacexby\frac{1}{x}$$$$\iff 2f\left(\frac{1}{x}\right)+f\left(x\right)=\frac{3}{x}$$$$\iff 4f\left(x\right)+2f\left(\frac{1}{x}\right)=6x$$$$\iff 2f\left(\frac{1}{x}\right)+f\left(x\right)=\frac{3}{x}$$$$\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}-$$$$\Rightarrow 3f\left(x\right)=6x-\frac{3}{x}$$$$\Rightarrow f\left(x\right)=2x-\frac{1}{x}=\frac{2x^2-1}{x}$$

Answered by mathmax by abdo last updated on 29/Aug/20

$$2\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)=3\mathrm{x}\mathrm{and}2\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)+\mathrm{f}\left(\mathrm{x}\right)=\frac{3}{\mathrm{x}}(\mathrm{x}\ne \mathrm{o})\mathrm{we}\mathrm{get}\mathrm{the}\mathrm{systeme}$$$$\{\begin{array}{l}2\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)=3\mathrm{x}\\ \mathrm{f}\left(\mathrm{x}\right)+2\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)=\frac{3}{\mathrm{x}}\end{array}$$$${\mathrm{\Delta }}_{\mathrm{s}}=\left|\begin{array}{c}21\\ 12\end{array}\right|=3\ne 0{\mathrm{\Delta }}_{\mathrm{f}\left(\mathrm{x}\right)}=\left|\begin{array}{c}3\mathrm{x}1\\ \frac{3}{\mathrm{x}}2\end{array}\right|=6\mathrm{x}-\frac{3}{\mathrm{x}}$$$${\mathrm{\Delta }}_{\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)}=\left|\begin{array}{c}23\mathrm{x}\\ 1\frac{3}{\mathrm{x}}\end{array}\right|=\frac{6}{\mathrm{x}}-3\mathrm{x}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\frac{{\mathrm{\Delta }}_{\mathrm{f}\left(\mathrm{x}\right)}}{\mathrm{\Delta }}=\frac{6\mathrm{x}-\frac{3}{\mathrm{x}}}{3}$$$$\mathrm{f}\left(\mathrm{x}\right)=2\mathrm{x}-\frac{1}{\mathrm{x}}$$

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