If f(x) + 2f(1/x) = 3x for all x ≠ 0, find f(x) — JEE Mathematics

The given functional equation is:

$$f(x) + 2f\left(\frac{1}{x}\right) = 3x \quad \dots (1)$$

Substitute $x$ with $\frac{1}{x}$ in equation (1):

$$f\left(\frac{1}{x}\right) + 2f(x) = \frac{3}{x} \quad \dots (2)$$

We want to eliminate $f(1/x)$. Multiply equation (2) by 2:

$$2f\left(\frac{1}{x}\right) + 4f(x) = \frac{6}{x} \quad \dots (3)$$

Subtract equation (1) from equation (3):

$$\left[2f\left(\frac{1}{x}\right) + 4f(x)\right] - \left[f(x) + 2f\left(\frac{1}{x}\right)\right] = \frac{6}{x} - 3x$$

$$3f(x) = \frac{6}{x} - 3x$$

Divide both sides by 3:

$$f(x) = \frac{2}{x} - x$$

Answer: $f(x) = \frac{2}{x} - x$