The function g is defined by \(\mathrm{g(x) = 6x}\). For what value of x is \(\mathrm{g(x) = 54}\)?

1. TRANSLATE the question into mathematical language

• Given information:
  • Function definition: \(\mathrm{g(x) = 6x}\)
  • We need: the value of x when \(\mathrm{g(x) = 54}\)

• What this tells us: We need to set \(\mathrm{g(x)}\) equal to 54 and solve for x

2. TRANSLATE to set up the equation

• Since \(\mathrm{g(x) = 6x}\) and we want \(\mathrm{g(x) = 54}\):
  • Substitute: \(\mathrm{54 = 6x}\)
• Now we have a simple linear equation to solve

3. SIMPLIFY to solve for x

• Divide both sides by 6:
  \(\mathrm{54 \div 6 = 6x \div 6}\)
  \(\mathrm{9 = x}\)

• Check our work: \(\mathrm{g(9) = 6(9) = 54}\) ✓

Answer: 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what the question is asking and try to substitute 54 for x instead of setting \(\mathrm{g(x)}\) equal to 54.

They might write \(\mathrm{g(54) = 6(54) = 324}\) and think this is the answer. This leads to confusion because 324 isn't typically an answer choice, causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equation correctly as \(\mathrm{54 = 6x}\), but make arithmetic errors when dividing.

Common calculation mistakes include \(\mathrm{54 \div 6 = 8}\) or \(\mathrm{54 \div 6 = 6}\), leading them to select incorrect answers if these values appear as choices.

The Bottom Line:

This problem tests whether students understand the fundamental relationship between function notation and equation solving. The key insight is recognizing that "\(\mathrm{g(x) = 54}\)" means we substitute the function definition and solve for the input value that produces the desired output.