The function g is defined by \(\mathrm{g(x) = 4x - 8}\). If \(\mathrm{g(a) = 12}\), what is the value of...

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{g(x) = 4x - 8}\)
  • Condition: \(\mathrm{g(a) = 12}\)
• What this tells us: We need to find the input value a that makes the output equal to 12

2. TRANSLATE the condition into an equation

• Since \(\mathrm{g(x) = 4x - 8}\), when the input is a:
  \(\mathrm{g(a) = 4a - 8}\)
• Setting this equal to the given output:
  \(\mathrm{4a - 8 = 12}\)

3. SIMPLIFY to solve for a

• Add 8 to both sides:
  \(\mathrm{4a - 8 + 8 = 12 + 8}\)
  \(\mathrm{4a = 20}\)
• Divide both sides by 4:
  \(\mathrm{a = 20 ÷ 4 = 5}\)

Answer: D (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not properly substitute a into the function definition, instead trying to work directly with \(\mathrm{g(a) = 12}\) without recognizing that \(\mathrm{g(a) = 4a - 8}\).

They might attempt to solve \(\mathrm{a = 12}\) directly or get confused about what the function notation means, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{4a - 8 = 12}\) but make arithmetic errors in the solving process, such as:

• Adding 8 incorrectly: \(\mathrm{4a = 12 - 8 = 4}\), leading to \(\mathrm{a = 1}\)
• Forgetting to add 8: solving \(\mathrm{4a = 12}\) directly, getting \(\mathrm{a = 3}\)

This may lead them to select Choice A (1) or Choice B (3).

The Bottom Line:

This problem tests whether students truly understand function notation as a substitution process and can execute basic equation-solving steps accurately. The key insight is recognizing that evaluating g(a) requires substituting a into the original function definition.