Question

If \(g(x) = 3x^2 + 2x - 3, f(0) = -3, 4g(f(x)) = 3x^2 - 32x + 72\) then find \(f(g(2))\) where \(f(x)>0\) for all valid x:

• A. \(7/2\)
• B. \(5/2\)
• C. \(3/2\)
• D. \(1/2\)

Hint:

When solving for a function inside a composition, look at the highest power of \(x\) on both sides. If \(g\) is quadratic and the result is quadratic, the inner function \(f\) must be linear.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem involves function composition and determining an unknown function \(f(x)\) by comparing coefficients.
Step 2: Key Formula or Approach:
Assume \(f(x)\) is a linear function \(ax + b\) based on the degrees of \(g(f(x))\) and the target polynomial.
Step 3: Detailed Explanation:
Given \(f(0) = -3\), let \(f(x) = px - 3\).
Now, \(g(f(x)) = 3(f(x))^2 + 2f(x) - 3\).
Given \(4g(f(x)) = 12(f(x))^2 + 8f(x) - 12 = 3x^2 - 32x + 72\).
Substitute \(f(x) = px - 3\):
\[ 12(px-3)^2 + 8(px-3) - 12 = 3x^2 - 32x + 72 \] \[ 12(p^2x^2 - 6px + 9) + 8px - 24 - 12 = 3x^2 - 32x + 72 \] \[ 12p^2x^2 - 72px + 108 + 8px - 36 = 3x^2 - 32x + 72 \] \[ 12p^2x^2 - 64px + 72 = 3x^2 - 32x + 72 \] Comparing coefficients:
1. \(12p^2 = 3 \implies p^2 = 1/4 \implies p = \pm 1/2\)
2. \(-64p = -32 \implies p = 1/2\)
So, \(f(x) = \frac{1}{2}x - 3\).
Now find \(f(g(2))\):
\(g(2) = 3(2)^2 + 2(2) - 3 = 12 + 4 - 3 = 13\).
\(f(g(2)) = f(13) = \frac{1}{2}(13) - 3 = 6.5 - 3 = 3.5 = \frac{7}{2}\).
Step 4: Final Answer:
The value of \(f(g(2))\) is \(7/2\).