Question:

Step 1: Write the given functional equation. $$2f(x) - 3f\left(\frac{1}{x}\right) = x^2 \quad \text{(Equation 1)}$$ Step 2: Substitute $x \to \frac{1}{x}$ in Equation 1 to obtain a second equation. $$2f\left(\frac{1}{x}\right) - 3f(x) = \frac{1}{x^2} \quad \text{(Equation 2)}$$ Step 3: Multiply Equation 1 by 2. $$4f(x) - 6f\left(\frac{1}{x}\right) = 2x^2 \quad \text{(Equation 1')}$$ Step 4: Multiply Equation 2 by 3. $$6f\left(\frac{1}{x}\right) - 9f(x) = \frac{3}{x^2} \quad \text{(Equation 2')}$$ Step 5: Add Equation 1' and Equation 2' to eliminate $f\left(\frac{1}{x}\right)$ and solve for $f(x)$. $$(4f(x) - 9f(x)) + \left(-6f\left(\frac{1}{x}\right) + 6f\left(\frac{1}{x}\right)\right) = 2x^2 + \frac{3}{x^2}$$ $$-5f(x) = 2x^2 + \frac{3}{x^2}$$ $$f(x) = -\frac{1}{5}\left(2x^2 + \frac{3}{x^2}\right) = -\frac{2x^4 + 3}{5x^2}$$ Step 6: Find $f(2)$ by substituting $x = 2$ into the expression for $f(x)$. $$f(2) = -\frac{2(2)^4 + 3}{5(2)^2} = -\frac{2(16) + 3}{5(4)} = -\frac{32 + 3}{20} = -\frac{35}{20} = -\frac{7}{4}$$ Step 7: Verify the result using the original equation at $x=2$. First, calculate $f\left(\frac{1}{2}\right)$: $$f\left(\frac{1}{2}\right) = -\frac{2\left(\frac{1}{2}\right)^4 + 3}{5\left(\frac{1}{2}\right)^2} = -\frac{2\left(\frac{1}{16}\right) + 3}{5\left(\frac{1}{4}\right)} = -\frac{\frac{1}{8} + 3}{\frac{5}{4}} = -\frac{\frac{1+24}{8}}{\frac{5}{4}} = -\frac{\frac{25}{8}}{\frac{5}{4}} = -\frac{25}{8} \cdot \frac{4}{5} = -\frac{5}{2}$$ Now, substitute $f(2)$ and $f\left(\frac{1}{2}\right)$ into Equation 1 for $x=2$: $$2f(2) - 3f\left(\frac{1}{2}\right) = 2\left(-\frac{7}{4}\right) - 3\left(-\frac{5}{2}\right) = -\frac{7}{2} + \frac{15}{2} = \frac{8}{2} = 4$$ The right-hand side of Equation 1 for $x=2$ is $x^2 = 2^2 = 4$. Since both sides are equal, the value of $f(2) = -\frac{7}{4}$ is correct.