The function f is defined by \(\mathrm{f(x) = 3x - 6}\). For what value of x is \(\mathrm{f(x) = 18}\)?

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{f(x) = 3x - 6}\)
  • Target condition: \(\mathrm{f(x) = 18}\)
• This means we need the function output to equal 18

2. TRANSLATE into algebraic equation

• Since \(\mathrm{f(x) = 3x - 6}\) and we want \(\mathrm{f(x) = 18}\):
• Set up: \(\mathrm{3x - 6 = 18}\)
• This gives us a linear equation to solve for x

3. SIMPLIFY to isolate the variable

• Add 6 to both sides: \(\mathrm{3x - 6 + 6 = 18 + 6}\)
• This gives us: \(\mathrm{3x = 24}\)
• Divide both sides by 3: \(\mathrm{x = 8}\)

4. Verify the solution

• Check: \(\mathrm{f(8) = 3(8) - 6 = 24 - 6 = 18}\) ✓

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might incorrectly set up the initial equation. Instead of recognizing that \(\mathrm{f(x) = 18}\) means "\(\mathrm{3x - 6 = 18}\)", they might write something like "\(\mathrm{3x - 6 + 18 = 0}\)" or get confused about what the equation should look like.

This leads to solving the wrong equation entirely, causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{3x - 6 = 18}\) but make arithmetic errors. They might add 6 incorrectly (getting \(\mathrm{3x = 23}\) instead of \(\mathrm{3x = 24}\)) or divide incorrectly at the final step.

These calculation mistakes lead to wrong numerical answers and confusion about whether their approach was correct.

The Bottom Line:

This problem tests whether students can bridge the gap between function notation and algebraic equation solving. The key insight is recognizing that "\(\mathrm{f(x) = 18}\)" is just another way of writing an equation that can be solved using standard algebraic techniques.