Question

If \[ g(x) = x^2 + x - 2 \] and \[ \frac{1}{2} g \circ f(x) = 2x^2 - 5x + 2, \] then \( f(x) \) is equal to:

• A. \( 2x - 3 \)
• B. \( 2x + 3 \)
• C. \( 2x^2 + 3x + 1 \)
• D. \( 2x^2 - 3x + 1 \)

Hint:

When solving functional equations, assume a possible form for the function (e.g., linear) and substitute it into the equation. Compare the coefficients of like powers of \( x \) to solve for the unknowns.

Answer 1

The Correct Option is A

We start with the equation for the composition of functions: \[ \frac{1}{2}g(f(x)) = 2x^2 - 5x + 2 \] This implies: \[ g(f(x)) = 4x^2 - 10x + 4 \] Assuming \( f(x) \) is quadratic, we get: \[ (f(x))^2 + f(x) - (4x^2 - 10x + 6) = 0 \] Solving this quadratic equation for \( f(x) \): \[ f(x) = \frac{-1 \pm \sqrt{1 + 4(4x^2 - 10x + 6)}}{2} \] \[ f(x) = \frac{-1 \pm \sqrt{16x^2 - 40x + 25}}{2} \] \[ f(x) = \frac{-1 \pm (4x - 5)}{2} \] Breaking this into two cases, we take the positive root: \[ f(x) = \frac{-1 + 4x - 5}{2} = 2x - 3 \] Thus, we find that: \[ f(x) = 2x - 3 \] Thus, the correct answer is Option A.