Question

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

• A. 3
• B. -3
• C. -2
• D. 2

Answer 1

Answer: (B)

-3

Answer 2

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.