For the function f, if \(\mathrm{f(3x) = x - 6}\) for all values of x, what is the value of...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(3x) = x - 6}\) for all values of x
• Find: \(\mathrm{f(6)}\)
• This means we need to figure out what happens when the input to function f is 6

2. INFER the key insight

• The given equation tells us what f does when its input is 3x
• To find \(\mathrm{f(6)}\), we need to determine what value of x makes \(\mathrm{3x = 6}\)
• Once we find that x-value, we can use it in the equation \(\mathrm{f(3x) = x - 6}\)

3. SIMPLIFY to find the required x-value

• Set up the equation: \(\mathrm{3x = 6}\)
• Divide both sides by 3: \(\mathrm{x = 2}\)

4. SIMPLIFY to evaluate f(6)

• Since \(\mathrm{x = 2}\) makes \(\mathrm{3x = 6}\), we substitute \(\mathrm{x = 2}\) into the right side:
• \(\mathrm{f(3·2) = 2 - 6}\)
• \(\mathrm{f(6) = -4}\)

Answer: B. -4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students don't recognize the connection between finding \(\mathrm{f(6)}\) and the given equation \(\mathrm{f(3x) = x - 6}\). They might directly substitute \(\mathrm{x = 6}\) into the right side of the equation, getting \(\mathrm{f(6) = 6 - 6 = 0}\).

This leads them to select Choice C (0).

Second Most Common Error:

Poor TRANSLATE interpretation: Students might confuse what they're looking for and think the answer should be the x-value itself rather than \(\mathrm{f(6)}\). Since \(\mathrm{x = 2}\) when \(\mathrm{3x = 6}\), they select the x-value as their final answer.

This may lead them to select Choice D (2).

The Bottom Line:

This problem tests whether students understand that function evaluation requires matching the input format. When given \(\mathrm{f(3x) = x - 6}\), finding \(\mathrm{f(6)}\) means finding what x makes \(\mathrm{3x = 6}\), then using that x in the equation.