Question:The function f is defined by \(\mathrm{f(x) = \frac{42}{x}}\). For what value of x does \(\mathrm{f(x) = 6}\)?

1. TRANSLATE the problem information

• Given information:
  • Function \(\mathrm{f(x) = \frac{42}{x}}\)
  • Need to find x where \(\mathrm{f(x) = 6}\)
• This means we need: \(\mathrm{\frac{42}{x} = 6}\)

2. SIMPLIFY to solve the equation

• Start with: \(\mathrm{\frac{42}{x} = 6}\)
• Multiply both sides by x: \(\mathrm{42 = 6x}\)
• Divide both sides by 6: \(\mathrm{x = \frac{42}{6} = 7}\)

Answer: B (7)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse the input and output of the function, thinking that since \(\mathrm{f(x) = 6}\), the answer must be \(\mathrm{x = 6}\).

They see "\(\mathrm{f(x) = 6}\)" and immediately think the variable x equals 6, without understanding that \(\mathrm{f(x)}\) represents the function's output, not its input. This leads them to select Choice A (6) without setting up the proper equation.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{\frac{42}{x} = 6}\) but make arithmetic errors when dividing 42 by 6.

They might calculate \(\mathrm{\frac{42}{6}}\) incorrectly, perhaps getting 14 (confusing it with \(\mathrm{\frac{42}{3}}\)) or making other computational mistakes. This could lead them to select Choice C (14) or cause confusion leading to guessing.

The Bottom Line:

This problem requires students to distinguish between a function's input and output, then execute straightforward algebraic manipulation. The key insight is recognizing that \(\mathrm{f(x) = 6}\) means "the function's output equals 6," not "x equals 6."