The function f is defined by \(\mathrm{f(x) = \frac{1}{2}(x + 6)}\). What is the value of \(\mathrm{f(4)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = \frac{1}{2}(x + 6)}\)
  • Need to find: \(\mathrm{f(4)}\)
• This means we need to substitute \(\mathrm{x = 4}\) into our function

2. SIMPLIFY by substituting and evaluating

• Substitute \(\mathrm{x = 4}\) into the function:

\(\mathrm{f(4) = \frac{1}{2}(4 + 6)}\)

• Follow order of operations - parentheses first:

\(\mathrm{f(4) = \frac{1}{2}(10)}\)

• Multiply:

\(\mathrm{f(4) = 5}\)

Answer: D. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misread the coefficient \(\mathrm{\frac{1}{2}}\) as 2, treating the function as \(\mathrm{f(x) = 2(x + 6)}\) instead of \(\mathrm{f(x) = \frac{1}{2}(x + 6)}\).

Following this incorrect reading:

\(\mathrm{f(4) = 2(4 + 6)}\)

\(\mathrm{= 2(10)}\)

\(\mathrm{= 20}\)

This may lead them to select Choice A (20).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret the multiplication structure, reading \(\mathrm{f(x) = \frac{1}{2}(x + 6)}\) as requiring addition rather than multiplication of the coefficient.

Following this logic:

\(\mathrm{f(4) = \frac{1}{2} + (4 + 6)}\)

\(\mathrm{= 0.5 + 10}\)

\(\mathrm{= 10.5}\)

then rounding or misreading to get 10, or completely ignoring the \(\mathrm{\frac{1}{2}}\) to get just \(\mathrm{(4 + 6) = 10}\).

This may lead them to select Choice C (10).

The Bottom Line:

This problem tests careful reading of mathematical notation and systematic function evaluation. Students who rush through the TRANSLATE step or don't pay close attention to coefficients and operation symbols will make predictable errors that align with the incorrect answer choices.