The function g is defined by \(\mathrm{g(x) = 3x - 12}\). If \(\mathrm{g(b) = 21}\), where b is a constant,...

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{g(x) = 3x - 12}\)
  • Function value: \(\mathrm{g(b) = 21}\)
  • Need to find: the value of b

2. TRANSLATE the function evaluation into an equation

• Since \(\mathrm{g(b) = 21}\), substitute b for x in the function definition:
• \(\mathrm{g(b) = 3b - 12 = 21}\)
• This gives us the equation: \(\mathrm{3b - 12 = 21}\)

3. SIMPLIFY by solving the linear equation

• Add 12 to both sides to isolate the term with b:
  \(\mathrm{3b - 12 + 12 = 21 + 12}\)
  \(\mathrm{3b = 33}\)

• Divide both sides by 3 to solve for b:
  \(\mathrm{b = 33 ÷ 3 = 11}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misunderstand function notation and think that \(\mathrm{g(b) = 21}\) means something other than substituting b into the function definition. They might try to work with \(\mathrm{g(x) = 3x - 12}\) and \(\mathrm{g(b) = 21}\) as separate, unrelated equations rather than recognizing that g(b) means "substitute b for x in the function definition."

This confusion leads to setting up incorrect equations or getting stuck, causing them to abandon systematic solution and guess.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(\mathrm{3b - 12 = 21}\) but make arithmetic errors. They might incorrectly calculate 21 + 12 = 32 instead of 33, leading to b = 32/3, which doesn't match any answer choice. Or they might make division errors with 33 ÷ 3.

This may lead them to select an incorrect choice or become confused and guess.

The Bottom Line:

This problem tests whether students understand function notation as a substitution process and can execute basic equation-solving steps accurately. The key insight is recognizing that \(\mathrm{g(b) = 21}\) creates a direct pathway to finding b through substitution.