The function f is defined by \(\mathrm{f(x) = x^2 + x + 71}\). What is the value of \(\mathrm{f(2)}\)?

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = x^2 + x + 71}\)
  • Need to find: \(\mathrm{f(2)}\)
• What this tells us: We need to substitute \(\mathrm{x = 2}\) into the function expression

2. SIMPLIFY by substituting and calculating

• Substitute \(\mathrm{x = 2}\) into \(\mathrm{f(x) = x^2 + x + 71}\):

\(\mathrm{f(2) = (2)^2 + 2 + 71}\)

• Calculate step by step using order of operations:
  • First, calculate the exponent: \(\mathrm{(2)^2 = 4}\)
  • Then add from left to right: \(\mathrm{4 + 2 + 71 = 77}\)

Answer: 77

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic calculation errors

Many students correctly understand they need to substitute \(\mathrm{x = 2}\), but then make mistakes like:

• Calculating \(\mathrm{(2)^2}\) as \(\mathrm{2}\) instead of \(\mathrm{4}\)
• Adding \(\mathrm{4 + 2 + 71}\) incorrectly (getting 76, 78, or other close values)

This leads to selecting an incorrect answer or confusion about their calculation.

Second Most Common Error:

Poor order of operations (SIMPLIFY): Students don't calculate the exponent first

Some students might calculate from left to right without following PEMDAS, treating the expression as:

\(\mathrm{2^2 + 2 + 71 = 2 + 2 + 2 + 71 = 77}\) (coincidentally still correct)

OR worse: \(\mathrm{2 \times 2 + 2 + 71}\) calculated as \(\mathrm{(2 \times 2 + 2) + 71 = 6 + 71 = 77}\) (again coincidentally correct)

However, with different numbers this approach would fail.

The Bottom Line:

This problem tests whether students understand function notation and can perform basic substitution with careful arithmetic - skills that seem simple but require attention to detail in execution.