\(\mathrm{f(x) = x^2 + 2x + 10}\) The function g is defined by \(\mathrm{g(x) = f(2x + 3)}\). For what...

1. INFER the strategic approach

• Key insight: Since \(\mathrm{g(x) = f(2x + 3)}\), the function g reaches its minimum when the input to f (which is \(\mathrm{2x + 3}\)) equals the x-value that minimizes f
• Strategy: First find where \(\mathrm{f(x)}\) is minimized, then set \(\mathrm{2x + 3}\) equal to that value

2. SIMPLIFY to find where f(x) reaches its minimum

• Complete the square for \(\mathrm{f(x) = x^2 + 2x + 10}\):

\(\mathrm{f(x) = x^2 + 2x + 1 - 1 + 10}\)

\(\mathrm{f(x) = (x + 1)^2 + 9}\)

• The minimum occurs at \(\mathrm{x = -1}\) (vertex of the parabola)

3. INFER the condition for g(x) to be minimized

• Since \(\mathrm{g(x) = f(2x + 3)}\), we need the input to f to equal -1
• This means: \(\mathrm{2x + 3 = -1}\)

4. SIMPLIFY to solve for x

\(\mathrm{2x + 3 = -1}\)

\(\mathrm{2x = -4}\)

\(\mathrm{x = -2}\)

Answer: C. -2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the composition structure and try to expand \(\mathrm{g(x)}\) completely first

They expand \(\mathrm{g(x) = f(2x + 3) = (2x + 3)^2 + 2(2x + 3) + 10}\), creating:

\(\mathrm{g(x) = 4x^2 + 12x + 9 + 4x + 6 + 10 = 4x^2 + 16x + 25}\)

Then they complete the square on this much more complicated expression, leading to calculation errors and confusion. This often causes them to get stuck and guess randomly.

Second Most Common Error:

Conceptual confusion about function composition: Students think \(\mathrm{g(x)}\) has its minimum at the same x-value as \(\mathrm{f(x)}\)

They find that \(\mathrm{f(x)}\) has its minimum at \(\mathrm{x = -1}\), then incorrectly conclude that \(\mathrm{g(x)}\) also has its minimum at \(\mathrm{x = -1}\). This may lead them to select Choice D (-1).

The Bottom Line:

The key breakthrough is recognizing that composition means finding when the "inner function" \(\mathrm{(2x + 3)}\) produces the optimal input for the "outer function" f. This insight transforms a potentially messy problem into two simple steps.