Question

If \( f(x) \) satisfies the relation \( 2f(x) + f(1 - x) = x^2 \) for all real \( x \), then \( f(x) \) is:

• A. \( x^2 + 2x - 1 \)
• B. \( x^2 + 2x - 1 \)
• C. \( x^2 + 4x - 1 \)
• D. \( x^2 + 4x - 1 \)

Hint:

To solve functional equations, substitute \( x \) and \( 1 - x \) to create a system of equations and solve for \( f(x) \).

Answer 1

The Correct Option is B

We are given the functional equation: \[ 2f(x) + f(1 - x) = x^2 \] Step 1: Substitute \( x = 1 - x \) Substitute \( 1 - x \) into the equation: \[ 2f(1 - x) + f(x) = (1 - x)^2 = 1 - 2x + x^2 \] Step 2: Solve the system of equations Now, we have the following system of equations: \[ 2f(x) + f(1 - x) = x^2 \quad \text{(1)} \] \[ 2f(1 - x) + f(x) = 1 - 2x + x^2 \quad \text{(2)} \] Solve these equations to get: \[ f(x) = x^2 + 2x - 1 \] Thus, the correct answer is \( f(x) = x^2 + 2x - 1 \).