The function g is defined by \(\mathrm{g(x) = x^2 + 9}\). For which value of x is \(\mathrm{g(x) = 25}\)?

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{g(x) = x^2 + 9}\)
  • We need: the value of x where \(\mathrm{g(x) = 25}\)

• This means we need to solve: \(\mathrm{x^2 + 9 = 25}\)

2. SIMPLIFY to isolate the variable term

• Subtract 9 from both sides:
  \(\mathrm{x^2 + 9 - 9 = 25 - 9}\)
  \(\mathrm{x^2 = 16}\)

3. SIMPLIFY further by taking the square root

• Take the square root of both sides:
  \(\mathrm{\sqrt{x^2} = \sqrt{16}}\)
  \(\mathrm{x = \pm 4}\)

4. CONSIDER ALL CASES and match with answer choices

• The equation gives us two solutions: \(\mathrm{x = 4}\) and \(\mathrm{x = -4}\)
• Looking at the choices [A. 4, B. 5, C. 9, D. 13], only 4 appears
• Therefore \(\mathrm{x = 4}\)

Answer: A. 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may substitute incorrectly or set up the wrong equation.

Some students might write \(\mathrm{g(25) = x^2 + 9}\) instead of setting \(\mathrm{g(x) = 25}\), confusing which variable should be substituted. This leads to trying to solve \(\mathrm{25 = x^2 + 9}\), which is actually the correct equation, but they may have arrived at it through faulty reasoning. However, this particular confusion wouldn't change the final answer.

More problematically, students might substitute 25 for x instead of g(x), writing \(\mathrm{g(25) = 25^2 + 9}\), which leads to a completely different problem. This leads to confusion and guessing.

Second Most Common Error:

Incomplete CONSIDER ALL CASES execution: Students solve correctly to get \(\mathrm{x^2 = 16}\) but only consider the positive square root.

While \(\mathrm{x = 4}\) is indeed the correct answer choice, students who automatically assume "square root means positive only" miss the complete mathematical picture. In this case, it doesn't affect the final answer since -4 isn't among the choices, but this incomplete understanding may hurt them on future problems. This may still lead them to select Choice A (4), but for incomplete mathematical reasons.

The Bottom Line:

This problem tests whether students can smoothly transition from function notation to algebraic equation solving while maintaining awareness of all mathematical solutions.