The function f is defined by the equation \(\mathrm{f(x) = 7x + 2}\). What is the value of \(\mathrm{f(x)}\) when...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = 7x + 2}\)
  • Need to find: \(\mathrm{f(x)}\) when \(\mathrm{x = 4}\)
• What this tells us: We need to substitute 4 for x in the function

2. TRANSLATE the substitution

• "Find \(\mathrm{f(x)}\) when \(\mathrm{x = 4}\)" means find \(\mathrm{f(4)}\)
• Replace every x in the function with 4:
  \(\mathrm{f(4) = 7(4) + 2}\)

3. SIMPLIFY the expression

• Follow order of operations (multiplication first):
  • \(\mathrm{7(4) = 28}\)
  • Then add: \(\mathrm{28 + 2 = 30}\)
• Therefore: \(\mathrm{f(4) = 30}\)

Answer: 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make basic arithmetic errors

Students might calculate \(\mathrm{7 \times 4}\) incorrectly (getting 32 instead of 28) or make addition errors. Even though this seems like a simple calculation, rushing or lack of attention to detail can lead to wrong answers like 34 instead of 30.

Second Most Common Error:

Poor TRANSLATE reasoning: Order of operations confusion

Some students might incorrectly think they should add first: \(\mathrm{f(4) = 7(4 + 2) = 7(6) = 42}\). This happens when they don't properly understand that the function is \(\mathrm{f(x) = 7x + 2}\), not \(\mathrm{f(x) = 7(x + 2)}\).

The Bottom Line:

This problem tests fundamental skills in function notation and arithmetic accuracy. Success requires careful substitution and methodical calculation following order of operations.