\(\mathrm{f(x) = 7x + 1}\). The function f gives the total number of people on a company retreat with x...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = 7x + 1}\)
  • This function gives the total number of people when there are x managers
  • We want the total number of people when there are 7 managers
• What this tells us: We need to find \(\mathrm{f(7)}\)

2. TRANSLATE what we need to find

• "Total number of people with 7 managers" means substitute \(\mathrm{x = 7}\) into our function
• So we're looking for \(\mathrm{f(7)}\)

3. SIMPLIFY by substituting and calculating

• \(\mathrm{f(7) = 7(7) + 1}\)
• \(\mathrm{f(7) = 49 + 1}\)
• \(\mathrm{f(7) = 50}\)

Answer: 50

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse what the variable x represents

Some students think x represents the total number of people instead of the number of managers. They might try to solve \(\mathrm{7x + 1 = 7}\) for x, thinking they need to find how many people total when there are 7 people. This backwards thinking leads to confusion and potentially guessing among answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic mistakes in basic multiplication

Students correctly identify they need \(\mathrm{f(7) = 7(7) + 1}\), but make calculation errors like \(\mathrm{7 × 7 = 48}\), leading them to get \(\mathrm{48 + 1 = 49}\) as their final answer instead of 50.

The Bottom Line:

This problem tests whether students can correctly interpret function notation in a real-world context. The key insight is recognizing that x (the input) represents managers, while f(x) (the output) represents total people - and not confusing these two roles.