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Question Number 108193 by bemath last updated on 15/Aug/20

$$\frac{\mathrm{♡}\mathcal{B}e\mathcal{M}ath\mathrm{♡}}{⧫}$$$$Given2f\left(x\right)+3f\left(\frac{1}{x}\right)=2x+3$$$$findf\left(3\right)?$$

Commented by bemath last updated on 15/Aug/20

$$thankyouboth$$

Answered by john santu last updated on 15/Aug/20

$$\frac{✓\mathcal{JS}✓}{\mathrm{♠}}$$$$putx=3\Rightarrow 2f\left(3\right)+3f\left(\frac{1}{3}\right)=9$$$$putx=\frac{1}{3}\Rightarrow 2f\left(\frac{1}{3}\right)+3f\left(3\right)=\frac{2}{3}+3$$$$\{\begin{array}{l}4f\left(3\right)+6f\left(\frac{1}{3}\right)=18\\ 9f\left(3\right)+6f\left(\frac{1}{3}\right)=11\end{array}$$$$\left(1\right)-\left(2\right)\Rightarrow -5f\left(3\right)=7$$$$\Rightarrow f\left(3\right)=-\frac{7}{5}$$

Answered by mr W last updated on 15/Aug/20

$$2f\left(x\right)+3f\left(\frac{1}{x}\right)=2x+3...\left(i\right)$$$$replacexwith\frac{1}{x}$$$$2f\left(\frac{1}{x}\right)+3f\left(x\right)=\frac{2}{x}+3...\left(ii\right)$$$$3\left(ii\right)-2\left(i\right):$$$$5f\left(x\right)=3+\frac{6}{x}-4x$$$$\Rightarrow f\left(x\right)=\frac{3}{5}+\frac{6}{5x}-\frac{4x}{5}$$$$\Rightarrow f\left(3\right)=-\frac{7}{5}$$

Answered by Dwaipayan Shikari last updated on 15/Aug/20

$$2f\left(3\right)+3f\left(\frac{1}{3}\right)=9\Rightarrow 4f\left(3\right)+6f\left(\frac{1}{3}\right)=18$$$$3f\left(3\right)+2f\left(\frac{1}{3}\right)=\frac{11}{3}\Rightarrow 9f\left(3\right)+6f\left(\frac{1}{3}\right)=11$$$$-5f\left(3\right)=7$$$$f\left(3\right)=-\frac{7}{5}$$$$$$

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