Question:The function g is defined by \(\mathrm{g(x) = 4x - 15}\). For what value of x does \(\mathrm{g(x) = 9}\)?

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{g(x) = 4x - 15}\)
  • We want to find: the value of x where \(\mathrm{g(x) = 9}\)
• What this tells us: We need to substitute 9 for \(\mathrm{g(x)}\) and solve for x

2. TRANSLATE the equation setup

• Since \(\mathrm{g(x) = 9}\) and \(\mathrm{g(x) = 4x - 15}\), we can write:
  \(\mathrm{4x - 15 = 9}\)

3. SIMPLIFY to solve for x

• Add 15 to both sides:
  \(\mathrm{4x - 15 + 15 = 9 + 15}\)
  \(\mathrm{4x = 24}\)

• Divide both sides by 4:
  \(\mathrm{x = 24 ÷ 4 = 6}\)

4. Verify the answer

• Check: \(\mathrm{g(6) = 4(6) - 15 = 24 - 15 = 9}\) ✓

Answer: B. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the input and output of the function, thinking that since \(\mathrm{g(x) = 9}\), they should substitute \(\mathrm{x = 9}\) into the function definition.

They calculate \(\mathrm{g(9) = 4(9) - 15 = 36 - 15 = 21}\), which doesn't help them find the answer. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equation correctly as \(\mathrm{4x - 15 = 9}\), but make arithmetic errors in the solving process.

Common mistakes include: \(\mathrm{9 + 15 = 25}\) (instead of 24) or \(\mathrm{24 ÷ 4 = 8}\) (calculation error). These errors may lead them to select incorrect answer choices or get frustrated and guess.

The Bottom Line:

This problem tests whether students understand that finding where \(\mathrm{g(x)}\) equals a specific value means setting the function definition equal to that value and solving for x, not plugging values into the function.