The function f is defined by \(\mathrm{f(x) = 4 + \sqrt{x}}\). What is the value of \(\mathrm{f(144)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = 4 + \sqrt{x}}\)
  • We need to find: \(\mathrm{f(144)}\)
• What this tells us: We need to substitute 144 for x in the function

2. TRANSLATE the substitution

• Replace x with 144 in the function:
  \(\mathrm{f(144) = 4 + \sqrt{144}}\)
• Now we need to evaluate this expression

3. SIMPLIFY the square root

• Calculate \(\mathrm{\sqrt{144}}\):
  Since \(\mathrm{12 \times 12 = 144}\), we have \(\mathrm{\sqrt{144} = 12}\)

4. SIMPLIFY the final arithmetic

• \(\mathrm{f(144) = 4 + 12 = 16}\)

Answer: B. 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Incorrectly calculating \(\mathrm{\sqrt{144}}\)

Students might not recognize that 144 is a perfect square, or they might miscalculate the square root. For example, they might think \(\mathrm{\sqrt{144} = 14}\) or struggle to find the exact value. This leads to wrong final answers and forces them to guess among the choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding function notation

Some students get confused about what \(\mathrm{f(144)}\) means and might try to solve \(\mathrm{f(x) = 144}\) instead of substituting 144 for x. This completely changes the problem and leads to confusion and random answer selection.

The Bottom Line:

This problem tests whether students can correctly substitute values into functions and work with square roots. The key insight is recognizing that function evaluation is just substitution followed by careful arithmetic.