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. \(2x^2+3x+2\)
• D. \(2x^2-3x-2\)

Hint:

Write g(f(x)) = f^2 + f - 2 and compare with the given polynomial.

Answer 1

The Correct Option is B

Step 1: Factor g:
$g(u)=u^2+u-2=(u+2)(u-1)$, so $g(f)=(f+2)(f-1)$.

Step 2: Factor the target:
$4x^2-10x+4=2(2x^2-5x+2)=2(2x-1)(x-2)=(2x-1)(2x-4)$.

Step 3: Match the factors:
If $f+2=2x-1$ then $f-1=2x-4$ is consistent (both give $f=2x-3$). So $f(x)=2x-3$, option B.

Final Answer:
Factoring g(f) gives (f + 2)(f - 1) = (2x - 1)(2x - 4), so f = 2x - 3. \[ \boxed{\text{(B) }2x-3} \]