\(\mathrm{f(x) = 14 + 4x}\). The function f represents the total cost, in dollars, of attending an arcade when x...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 14 + 4x}\) (total cost function)
  • Need total cost of $58
• This tells us we need \(\mathrm{f(x) = 58}\)

2. INFER the solution approach

• Since we know the function and the desired output, we substitute
• Set up the equation: \(\mathrm{58 = 14 + 4x}\)
• Now we solve for x using algebraic steps

3. SIMPLIFY to isolate x

• Start with: \(\mathrm{58 = 14 + 4x}\)
• Subtract 14 from both sides: \(\mathrm{58 - 14 = 4x}\)
• This gives us: \(\mathrm{44 = 4x}\)
• Divide both sides by 4: \(\mathrm{44 \div 4 = x}\)
• Therefore: \(\mathrm{x = 11}\)

Answer: 11 games

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand that 'total cost of $58' means setting \(\mathrm{f(x) = 58}\). They might try to work backwards from the function format or get confused about what the question is asking.

This leads to confusion and random guessing rather than systematic equation setup.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{58 = 14 + 4x}\) but make arithmetic errors. For example, they might subtract incorrectly (\(\mathrm{58 - 14 = 54}\)) or divide incorrectly (\(\mathrm{44 \div 4 = 12}\)).

These calculation errors lead to wrong final answers like 12 or 13.5 games.

The Bottom Line:

This problem tests whether students can connect function notation to real-world scenarios and then execute basic equation solving. The key insight is recognizing that finding 'how many games for $58' means setting the function equal to 58.