Let \(\mathrm{f(x) = 3x^2 - 18x + 41}\). Define \(\mathrm{g(x) = f(2x - 5)}\). For what value of x does...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 3x^2 - 18x + 41}\)
  • \(\mathrm{g(x) = f(2x - 5)}\)
  • Need to find x where g(x) is minimized

2. INFER the key relationship

• Since \(\mathrm{g(x) = f(2x - 5)}\), we're evaluating f at the point \(\mathrm{(2x - 5)}\)
• g(x) will be minimized when the input to f, which is \(\mathrm{(2x - 5)}\), equals the x-value where f itself is minimized
• Strategy: Find where f(x) is minimized, then solve \(\mathrm{2x - 5}\) = that value

3. SIMPLIFY to find f's minimum point

• For \(\mathrm{f(x) = 3x^2 - 18x + 41}\), use vertex formula \(\mathrm{x = -b/(2a)}\)
• Here \(\mathrm{a = 3, b = -18}\)
• Minimum occurs at: \(\mathrm{x = -(-18)/(2 \cdot 3) = 18/6 = 3}\)

4. SIMPLIFY to find where g is minimized

• Set the input to f equal to f's minimizer: \(\mathrm{2x - 5 = 3}\)
• Solve: \(\mathrm{2x = 8}\)
• Therefore: \(\mathrm{x = 4}\)

Answer: C (4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the composition relationship and try to find the minimum of f(x) directly, then select \(\mathrm{x = 3}\) as their answer.

They think: "f(x) is minimized at \(\mathrm{x = 3}\), so that must be the answer." This fundamental misunderstanding of what g(x) represents leads them to select Choice B (3).

Second Most Common Error:

Poor TRANSLATE reasoning: Students attempt to expand \(\mathrm{g(x) = f(2x - 5)}\) by substituting, but make algebraic errors in the expansion process.

For example, they might incorrectly expand to get \(\mathrm{g(x) = 3(2x - 5)^2 - 18(2x - 5) + 41}\) but then make calculation mistakes, leading to wrong coefficients and an incorrect vertex calculation. This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students understand function composition and can distinguish between minimizing the original function versus minimizing the composite function. The key insight is recognizing that g(x) inherits its minimum from f(x), but at a transformed input value.