Question:The function g is defined by \(\mathrm{g(x) = 3x}\). For what value of x does \(\mathrm{g(x) = 12}\)?Enter your answer...

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{g(x) = 3x}\)
  • Question: "For what value of x does \(\mathrm{g(x) = 12}\)?"
• What this tells us: We need to find the x-value that makes the function output equal to 12

2. TRANSLATE the question into an equation

• The question "For what value of x does \(\mathrm{g(x) = 12}\)?" means we need to solve:
  \(\mathrm{g(x) = 12}\)
• Since \(\mathrm{g(x) = 3x}\), we substitute to get:
  \(\mathrm{3x = 12}\)

3. SIMPLIFY by solving the equation

• We have: \(\mathrm{3x = 12}\)
• Divide both sides by 3:
  \(\mathrm{x = 12 ÷ 3 = 4}\)
• Check our answer:
  \(\mathrm{g(4) = 3(4) = 12}\) ✓

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand what the question is asking and fail to set up the correct equation.

They might think the question is asking for the value of g(x) when x = 12, leading them to calculate \(\mathrm{g(12) = 3(12) = 36}\) instead of solving for when \(\mathrm{g(x) = 12}\). This leads to confusion and guessing since 36 isn't the format requested (integer answer for x).

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equation correctly as \(\mathrm{3x = 12}\) but make arithmetic errors.

They might incorrectly compute \(\mathrm{12 ÷ 3}\), perhaps getting 3 or some other wrong value due to rushed calculation or basic arithmetic confusion. This causes them to submit an incorrect integer answer.

The Bottom Line:

This problem tests whether students can bridge the gap between function notation and equation solving. The key insight is recognizing that "\(\mathrm{g(x) = 12}\)" is asking you to find the input that produces output 12, not asking you to evaluate the function at x = 12.