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

Step 1: Understanding the Concept:
This problem involves composite functions and functional equations.
We are given the structure of \(g(x)\) and a condition for \(4g(f(x))\).
The goal is to find the function \(f(x)\) and then evaluate the composition \(f(g(2))\).
Step 2: Key Formula or Approach:
Since \(g(x)\) is a quadratic function and \(g(f(x))\) is also a quadratic function, \(f(x)\) must be a linear function.
Let \(f(x) = ax + b\).
Step 3: Detailed Explanation:
Given \(f(0) = -3\), substituting this into our linear assumption:
\(f(0) = a(0) + b = -3 \Rightarrow b = -3\).
Thus, \(f(x) = ax - 3\).
Now, substitute \(f(x)\) into \(g(x)\):
\(g(f(x)) = 3(ax - 3)^2 + 2(ax - 3) - 3\).
Expanding the expression:
\(g(f(x)) = 3(a^2x^2 - 6ax + 9) + 2ax - 6 - 3\).
\(g(f(x)) = 3a^2x^2 - 18ax + 27 + 2ax - 9\).
\(g(f(x)) = 3a^2x^2 - 16ax + 18\).
Multiplying by 4 as per the given condition:
\(4g(f(x)) = 12a^2x^2 - 64ax + 72\).
Comparing this with the given expression \(3x^2 - 32x + 72\):
For \(x^2\) coefficients: \(12a^2 = 3 \Rightarrow a^2 = \frac{1}{4} \Rightarrow a = \pm \frac{1}{2}\).
For \(x\) coefficients: \(-64a = -32 \Rightarrow a = \frac{1}{2}\).
So, \(f(x) = \frac{1}{2}x - 3\).
Now, calculate \(g(2)\):
\(g(2) = 3(2)^2 + 2(2) - 3 = 12 + 4 - 3 = 13\).
Finally, find \(f(g(2)) = f(13)\):
\(f(13) = \frac{1}{2}(13) - 3 = 6.5 - 3 = 3.5 = \frac{7}{2}\).
Step 4: Final Answer:
The value of \(f(g(2))\) is \(\frac{7}{2}\).