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

To solve the problem, we need to determine \(f(g(2))\) based on the given information and constraints.

1. Calculate \(g(2)\): 

The function \(g(x)\) is defined as:

1. \(g(x) = 3x^2 + 2x - 3\)

Substituting \(x = 2\) into \(g(x)\):

1. \(g(2) = 3(2)^2 + 2(2) - 3\) \(= 3 \cdot 4 + 4 - 3 = 12 + 4 - 3 = 13\)

Thus, \(g(2) = 13\).

1. Understand the function \(f(x)\):

We have the condition:

1. \(4g(f(x)) = 3x^2 - 32x + 72\)

Since \(g(x) = 3x^2 + 2x - 3\), replace \(g(f(x))\) with its expression:

1. \(4(3f(x)^2 + 2f(x) - 3) = 3x^2 - 32x + 72\) \(12f(x)^2 + 8f(x) - 12 = 3x^2 - 32x + 72\)

Dividing the equation by 4, we get:

1. \(3f(x)^2 + 2f(x) - 3 = \frac{3x^2 - 32x + 72}{4}\)

Find \(f(g(2))\):

Since \(g(2) = 13\), we need to find \(f(13)\).

Using the derived equation:

1. \(3f(13)^2 + 2f(13) - 3 = \frac{3(13)^2 - 32(13) + 72}{4}\)

Calculate for the right side:

1. \(3(169) = 507; -32 \times 13 = -416; 507 - 416 + 72 = 163; \frac{163}{4} = 40.75\)

Thus, solve for \(f(x)\) in:

1. \(3f(13)^2 + 2f(13) - 3 = 40.75\)

Solving this quadratic:

1. \(3t^2 + 2t - 43.75 = 0\)

Using the quadratic formula:

1. \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) \(t = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 3 \cdot (-43.75)}}{2 \cdot 3}\) \(= \frac{-2 \pm \sqrt{4 + 525}}{6}\) \(= \frac{-2 \pm \sqrt{529}}{6}\) \(= \frac{-2 \pm 23}{6}\)

Thus, two solutions:

1. \(t = \frac{21}{6} = \frac{7}{2} \, \text{and} \, t = -\frac{25}{6}\)

Since \(f(x) > 0\), we take the positive solution:

Thus, \(f(g(2)) = \frac{7}{2}\).

The correct answer is \(\frac{7}{2}\).