The function g is defined by \(\mathrm{g(x) = -3x + 27}\). For what value of x is \(\mathrm{g(x) = 42}\)?

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = -3x + 27}\)
  • We need \(\mathrm{g(x) = 42}\)
• What this tells us: We need to substitute 42 for g(x) in our function

2. TRANSLATE to set up the equation

• Since \(\mathrm{g(x) = -3x + 27}\) and we want \(\mathrm{g(x) = 42}\):
• This gives us: \(\mathrm{-3x + 27 = 42}\)

3. SIMPLIFY by isolating the variable term

• Subtract 27 from both sides: \(\mathrm{-3x + 27 - 27 = 42 - 27}\)
• This gives us: \(\mathrm{-3x = 15}\)

4. SIMPLIFY to solve for x

• Divide both sides by -3: \(\mathrm{x = 15 ÷ (-3)}\)
• Calculating: \(\mathrm{x = -5}\)

Answer: B) -5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign confusion when dividing by negative numbers

Students correctly set up \(\mathrm{-3x = 15}\), but then make the error: \(\mathrm{x = 15/3 = 5}\), forgetting that they're dividing by -3, not +3. The negative sign is easy to overlook in the final step.

This may lead them to select Choice D (5).

Second Most Common Error:

Poor TRANSLATE reasoning: Confusion about function notation

Some students might think \(\mathrm{g(x) = 42}\) means \(\mathrm{x = 42}\), completely bypassing the function definition. They don't realize they need to substitute 42 into the original function equation.

This leads to confusion and guessing among the available choices.

The Bottom Line:

This problem tests whether students can work systematically with function notation and maintain careful attention to signs during algebraic manipulation. The function notation is straightforward, but the negative coefficient requires extra care in the final division step.