Functions Part 2

Concept: This problem involves function composition and determining the unknown function $f(x)$ by matching the coefficients of the given composite function $g(f(x))$ with the known structure of $g(x)$.Formula: Function composition: $g(f(x))$ means substituting $f(x)$ into the variable $x$ of the function $g(x)$.Solution:Step 1: Calculate $g(2)$.Given $g(x) = 3x^2 + 2x - 3$.$g(2) = 3(2)^2 + 2(2) - 3$$g(2) = 3(4) + 4 - 3$$g(2) = 12 + 1 = 13$We need to find $f(g(2)) = f(13)$.Step 2: Determine the function $f(x)$.We are given $4g(f(x)) = 3x^2 - 32x + 72$.$g(f(x)) = \;\frac{1}{4}(3x^2 - 32x + 72)$$g(f(x)) = \;\frac{3}{4}x^2 - 8x + 18$Since $g(x)$ is a quadratic function, and $g(f(x))$ is a quadratic function, we assume $f(x)$ is a linear function: $f(x) = ax + b$.Using the initial condition $f(0) = -3$:$f(0) = a(0) + b = -3 \Rightarrow b = -3$Thus, $f(x) = ax - 3$.Now substitute $f(x)$ into $g(x)$:$g(f(x)) = 3(ax - 3)^2 + 2(ax - 3) - 3$$g(f(x)) = 3(a^2x^2 - 6ax + 9) + 2ax - 6 - 3$$g(f(x)) = 3a^2x^2 - 18ax + 27 + 2ax - 9$$g(f(x)) = (3a^2)x^2 + (-18a + 2a)x + (18)$$g(f(x)) = (3a^2)x^2 + (-16a)x + 18$Step 3: Find the coefficient $a$ by comparing coefficients.We equate the derived expression for $g(f(x))$ with the given expression:$(3a^2)x^2 + (-16a)x + 18 = \;\frac{3}{4}x^2 - 8x + 18$Comparing the coefficients of $x^2$:$3a^2 = \;\frac{3}{4} \Rightarrow a^2 = \;\frac{1}{4} \Rightarrow a = \pm \frac{1}{2}$Comparing the coefficients of $x$:$-16a = -8 \Rightarrow a = \;\frac{-8}{-16} = \;\frac{1}{2}$The value $a = \frac{1}{2}$ satisfies both conditions.Therefore, the function $f(x)$ is:$f(x) = \;\frac{1}{2}x - 3$Step 4: Calculate $f(13)$.$f(13) = \;\frac{1}{2}(13) - 3$$f(13) = \;\frac{13}{2} - \;\frac{6}{2}$$f(13) = \;\frac{7}{2}$The value of $f(g(2))$ is $\;\frac{7}{2}$.***Answer: A. $\;\frac{7}{2}$