For any non-zero real number x, let\( f(x) + 2f\left(\frac{1}{x}\right) = 3x.\)Then, the sum of all possible values of x for which f(x) = 3, is
- 3
- -3
- -2
- 2
The Correct Option is B
Solution and Explanation
We are given the functional equation:
\[ f(x) + 2f\left(\frac{1}{x}\right) = 3x \]
We are asked to find the sum of all possible values of $x$ for which $f(x) = 3$.
Substitute $f(x) = 3$ into the equation:
\[ 3 + 2f\left(\frac{1}{x}\right) = 3x \]
Solve for $f\left(\frac{1}{x}\right)$:
\[ 2f\left(\frac{1}{x}\right) = 3x - 3 \]
\[ f\left(\frac{1}{x}\right) = \frac{3x - 3}{2} \]
Now, substitute $x = \frac{1}{x}$ into the original equation:
\[ f\left(\frac{1}{x}\right) + 2f(x) = \frac{3}{x} \]
This results in a system of equations, which can be solved to find the value of $x$. After solving the system, we find that the sum of all possible values of $x$ for which $f(x) = 3$ is -3.
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