Question

If \( g(x) = 3x^2 + 2x - 3 \), \( f(0) = -3 \) and \( 4g(f(x)) = 3x^2 - 32x + 72 \), then \( f(g(2)) \) is equal to:

• A. \( -\dfrac{25}{6} \)
• B. \( -\dfrac{7}{2} \)
• C. \( \dfrac{25}{6} \)
• D. \( \dfrac{7}{2} \)

Hint:

When a composite function is given, first rewrite it in simplified form and compare coefficients to identify the unknown function.

Answer 1

The Correct Option is D

Concept: If a function is quadratic and composed with a linear function, then comparing coefficients allows us to determine the unknown function. Once the functions are known, direct substitution gives the required value.
Step 1: Use the given information about \( g(f(x)) \) Given: \[ 4g(f(x)) = 3x^2 - 32x + 72 \] \[ \Rightarrow g(f(x)) = \frac{3}{4}x^2 - 8x + 18 \]
Step 2: Assume \( f(x) \) is a linear function Let: \[ f(x) = ax + b \] Given \( f(0) = -3 \Rightarrow b = -3 \) So, \[ f(x) = ax - 3 \]
Step 3: Substitute \( f(x) \) into \( g(x) \) \[ g(f(x)) = 3(ax-3)^2 + 2(ax-3) - 3 \] \[ = 3a^2x^2 - 18ax + 27 + 2ax - 6 - 3 \] \[ = 3a^2x^2 - 16ax + 18 \]
Step 4: Compare coefficients \[ 3a^2 = \frac{3}{4} \Rightarrow a^2 = \frac{1}{4} \] \[ -16a = -8 \Rightarrow a = \frac{1}{2} \] Thus, \[ f(x) = \frac{x}{2} - 3 \]
Step 5: Find \( g(2) \) \[ g(2) = 3(2)^2 + 2(2) - 3 = 12 + 4 - 3 = 13 \]
Step 6: Find \( f(g(2)) \) \[ f(13) = \frac{13}{2} - 3 = \frac{7}{2} \]