For the function f, \(\mathrm{f(cx) = x - 8}\) for all values of x, where c is a positive constant....

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(cx) = x - 8}\) for all values of x
  • c is a positive constant
  • \(\mathrm{f(2) = 35}\)
• What this tells us: We need to find what value of c makes this function relationship work

2. INFER the key insight

• To evaluate f(2), we need to find what input value x makes \(\mathrm{cx = 2}\)
• This means we need \(\mathrm{cx = 2}\), so \(\mathrm{x = \frac{2}{c}}\)
• This x value is what we substitute into the function definition

3. TRANSLATE the function evaluation

• Since \(\mathrm{f(cx) = x - 8}\) and we want f(2):
• \(\mathrm{f(2) = x - 8}\), where \(\mathrm{cx = 2}\)
• Substituting \(\mathrm{x = \frac{2}{c}}\): \(\mathrm{f(2) = \frac{2}{c} - 8}\)

4. SIMPLIFY using the given condition

• We know \(\mathrm{f(2) = 35}\), so:

\(\mathrm{35 = \frac{2}{c} - 8}\)

• Add 8 to both sides:

\(\mathrm{43 = \frac{2}{c}}\)

• Multiply both sides by c:

\(\mathrm{43c = 2}\)

• Divide by 43:

\(\mathrm{c = \frac{2}{43}}\)

5. Convert to decimal form

• \(\mathrm{c = \frac{2}{43} = 0.0465}\) (use calculator)

Answer: 2/43 or 0.0465

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often misunderstand what f(2) means in the context of \(\mathrm{f(cx) = x - 8}\). They might try to substitute \(\mathrm{x = 2}\) directly into the function, getting \(\mathrm{f(2c) = 2 - 8 = -6}\), then incorrectly set this equal to 35. This leads to confusion because they're evaluating the wrong input, and they typically get stuck and resort to guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Even when students correctly set up \(\mathrm{35 = \frac{2}{c} - 8}\), they often make algebraic errors in the manipulation. Common mistakes include forgetting to add 8 to both sides, or incorrectly inverting the fraction relationship. This may lead them to calculate an incorrect value like \(\mathrm{c = \frac{43}{2} = 21.5}\), which would correspond to a decimal around 0.047 if they're working backwards.

The Bottom Line:

This problem requires recognizing that function notation f(2) means "what happens when the INPUT to function f equals 2," not "what happens when x = 2 in the function definition." The key insight is understanding the relationship between the function's input and the parameter in f(cx).