The function f is defined by \(\mathrm{f(x) = 4x}\). For what value of x does \(\mathrm{f(x) = 8}\)?

1. TRANSLATE the problem information

• Given information:
  • The function is defined as \(\mathrm{f(x) = 4x}\)
  • We need to find x when \(\mathrm{f(x) = 8}\)
• This tells us we need to solve: \(\mathrm{4x = 8}\)

2. SIMPLIFY to solve the equation

• We have the equation: \(\mathrm{4x = 8}\)
• Divide both sides by 4: \(\mathrm{x = 8/4 = 2}\)
• Check: \(\mathrm{f(2) = 4(2) = 8}\) ✓

Answer: 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not understand how to convert the function question into an equation. They might see \(\mathrm{f(x) = 8}\) and \(\mathrm{f(x) = 4x}\), but struggle to connect these to form \(\mathrm{8 = 4x}\).

Some students might incorrectly set up equations like:

• \(\mathrm{4 = 8x}\) (confusing which expression equals 8)
• \(\mathrm{f(8) = 4x}\) (misinterpreting what the question is asking)

This leads to confusion and incorrect solutions, or causes them to get stuck and guess.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(\mathrm{8 = 4x}\) but make arithmetic mistakes when solving, such as:

• Incorrectly calculating \(\mathrm{8 ÷ 4}\)
• Dividing the wrong way (getting \(\mathrm{x = 4/8 = 1/2}\))
• Adding or subtracting instead of dividing

This leads them to arrive at incorrect numerical answers.

The Bottom Line:

This problem tests whether students can bridge the gap between function notation and basic equation solving. The key challenge is recognizing that "\(\mathrm{f(x) = 8}\)" combined with "\(\mathrm{f(x) = 4x}\)" means "\(\mathrm{4x = 8}\)."