The function f is defined by \(\mathrm{f(x) = \frac{1}{6x}}\). What is the value of \(\mathrm{f(x)}\) when x = 3?

1. TRANSLATE the problem information

• Given information:
  • Function definition: \(\mathrm{f(x) = \frac{1}{6x}}\)
  • Need to find: \(\mathrm{f(x)}\) when \(\mathrm{x = 3}\)
• This means we need to find \(\mathrm{f(3)}\) by substituting 3 for x in the function

2. SIMPLIFY by substitution and calculation

• Substitute \(\mathrm{x = 3}\) into \(\mathrm{f(x) = \frac{1}{6x}}\):
  \(\mathrm{f(3) = \frac{1}{6(3)}}\)
• Calculate the denominator: \(\mathrm{6 \times 3 = 18}\)
• Therefore: \(\mathrm{f(3) = \frac{1}{18}}\)

Answer: D. \(\mathrm{\frac{1}{18}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret the function notation or the substitution requirement.

Some students might think \(\mathrm{f(x) = \frac{1}{6x}}\) means \(\mathrm{f(x) = (\frac{1}{6})x}\), leading them to calculate \(\mathrm{(\frac{1}{6})(3) = \frac{1}{2}}\), which isn't even among the choices. This causes confusion and may lead to guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors in basic calculations.

A student might correctly set up \(\mathrm{f(3) = \frac{1}{6(3)}}\) but then calculate \(\mathrm{6 \times 3}\) incorrectly (perhaps getting 9 instead of 18), leading them to \(\mathrm{f(3) = \frac{1}{9}}\). This may lead them to select Choice C (\(\mathrm{\frac{1}{9}}\)).

The Bottom Line:

This problem tests whether students can correctly interpret function notation and perform accurate substitution. The key insight is recognizing that "\(\mathrm{f(x)}\) when \(\mathrm{x = 3}\)" simply means "replace every x with 3 in the function definition."