+26367 What is \(f(2010)\) if \(3f(x)-5xf\left(\dfrac{1}{x}\right)=x-7\)
\(\begin{array}{|lrcll|} \hline & \mathbf{3f(x) - 5xf\left(\dfrac{1}{x}\right)} &=& \mathbf{x-7} \qquad (1) \\ x\rightarrow \dfrac{1}{x}: & \mathbf{3f\left(\dfrac{1}{x}\right) - \dfrac{5}{x}f(x)} &=& \mathbf{\dfrac{1}{x}-7} \qquad (2) \\ & \boxed{\text{Let } a=f(x), \ b=f\left(\dfrac{1}{x}\right) } \\ & \mathbf{3a - 5xb} &=& \mathbf{x-7} \qquad (1) \\ & \mathbf{3b - \dfrac{5}{x}a} &=& \mathbf{\dfrac{1}{x}-7} \qquad (2) \\ \hline \end{array} \)
\(\begin{array}{|lrcll|} \hline (2): & \mathbf{3b - \dfrac{5}{x}a} &=& \mathbf{\dfrac{1}{x}-7} \\\\ & 3b &=& \dfrac{1}{x}-7 + \dfrac{5}{x}a \\\\ & \mathbf{b} &=& \mathbf{\dfrac{ \dfrac{1}{x}-7 + \dfrac{5}{x}a } {3}} \\ \hline \end{array} \begin{array}{|lrcll|} \hline (1): & \mathbf{3a - 5xb} &=& \mathbf{x-7} \\\\ & 3a - 5x\left(\dfrac{ \dfrac{1}{x}-7 + \dfrac{5}{x}a } {3}\right) &=& x-7 \quad | \quad *3 \\\\ & 9a - 5x\left( \dfrac{1}{x}-7 + \dfrac{5}{x}a \right) &=& 3x - 21 \\\\ & 9a - 5 + 35x + 25a &=& 3x - 21 \\\\ & -16a+32x &=& -16 \quad | \quad :(-16) \\\\ & a-2x &=& 1 \\\\ & \mathbf{a} &=& \mathbf{2x+1} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \mathbf{f(x)} &=& \mathbf{2x+1} \quad | \quad x = 2010 \\\\ f(2010) &=& 2*2010+1 \\ \mathbf{f(2010)} &=& \mathbf{4021} \\ \hline \end{array}\)
