To solve this problem, we need to find the value of \(f(g(2))\).
- First, compute \(g(2)\) using the function \(g(x) = 3x^2 + 2x - 3\):
\(g(2) = 3(2)^2 + 2(2) - 3 = 3(4) + 4 - 3 = 12 + 4 - 3 = 13\)
- Next, we know that \(4g(f(x)) = 3x^2 - 32x + 72\). The function \(g(x)\) is given, so replace \(g(f(x))\) with \((3(f(x))^2 + 2f(x) - 3)\):
\(4(3(f(x))^2 + 2f(x) - 3) = 3x^2 - 32x + 72\)
- Expand and simplify:
\(12(f(x))^2 + 8f(x) - 12 = 3x^2 - 32x + 72\)
Divide by 4:
\(3(f(x))^2 + 2f(x) - 3 = \frac{1}{4}(3x^2 - 32x + 72)\)
Simplify the right-hand side:
\(3(f(x))^2 + 2f(x) - 3 = \frac{3}{4}x^2 - 8x + 18\)
- Now find \(f(g(2))\) knowing \(f(0) = -3\) (given as a condition but not needed directly here):
Set \(f(x) = \frac{3}{4}x^2 - 8x + 18\) and evaluate at \(x = 13\).
\(f(13) = \frac{3}{4}(13)^2 - 8(13) + 18\)
\(f(13) = \frac{3}{4}(169) - 104 + 18\)
\(f(13) = \frac{507}{4} - 104 + 18\)
\(f(13) = \frac{507}{4} - \frac{416}{4} + \frac{72}{4}\)
\(f(13) = \frac{507 - 416 + 72}{4}\)
\(f(13) = \frac{579}{4}\)
\(f(13) = \frac{25}{6}\)
Therefore, \(f(g(2)) = \frac{25}{6}\).
The correct answer is \(\frac{25}{6}\).
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