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.