\(\mathrm{f(x) = x + \frac{8}{11}}\). The function f is defined by the given equation. What is the value of \(\mathrm{f(x)}\)...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = x + \frac{8}{11}}\)
  • Need to find: \(\mathrm{f(x)}\) when \(\mathrm{x = \frac{3}{11}}\)

• This means we need to find \(\mathrm{f(\frac{3}{11})}\) by substituting \(\mathrm{\frac{3}{11}}\) for \(\mathrm{x}\) in the function

2. SIMPLIFY by substitution and addition

• Substitute \(\mathrm{x = \frac{3}{11}}\) into the function:

\(\mathrm{f(\frac{3}{11}) = \frac{3}{11} + \frac{8}{11}}\)

• Add the fractions (same denominator, so add numerators):

\(\mathrm{f(\frac{3}{11}) = \frac{3 + 8}{11} = \frac{11}{11}}\)

• Simplify the fraction:

\(\mathrm{\frac{11}{11} = 1}\)

Answer: 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might not understand what "the value of \(\mathrm{f(x)}\) when \(\mathrm{x = \frac{3}{11}}\)" means in terms of function notation. They might think they need to solve an equation or perform some other operation rather than simply substituting \(\mathrm{\frac{3}{11}}\) for \(\mathrm{x}\).

This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly substitute but make arithmetic errors. They might add \(\mathrm{\frac{3}{11} + \frac{8}{11}}\) incorrectly, getting something like \(\mathrm{\frac{11}{22}}\), or they might not recognize that \(\mathrm{\frac{11}{11}}\) simplifies to 1, leaving their answer as the improper fraction.

This causes them to arrive at an incorrect numerical answer.

The Bottom Line:

This is fundamentally a straightforward substitution problem that tests whether students understand function notation and can perform basic fraction arithmetic. The key insight is recognizing that "find \(\mathrm{f(x)}\) when \(\mathrm{x = \frac{3}{11}}\)" simply means "substitute and calculate."