To repair a refrigerator, a technician charges $60 per hour for labor plus $120 for parts. Which function f represents...

1. TRANSLATE the problem information

• Given information:
  • \(\$60\) per hour for labor (this varies with time)
  • \(\$120\) for parts (this is fixed, regardless of time)
  • x represents hours worked

2. INFER the cost structure

• Total cost has two components:
  • Variable cost that depends on hours: Labor
  • Fixed cost that doesn't change: Parts
• We need to express each mathematically, then combine them

3. Calculate the variable cost component

• Labor cost = (rate per hour) × (number of hours)
• Labor cost = \(60 \times x = 60x\) dollars

4. Identify the fixed cost component

• Parts cost = \(\$120\) (doesn't matter if job takes 1 hour or 10 hours)

5. TRANSLATE to create the total cost function

• Total cost = Variable cost + Fixed cost
• \(f(x) = 60x + 120\)

Answer: C. \(f(x) = 60x + 120\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly interpret "per hour" language and create \(f(x) = x + 120\), forgetting that the \(\$60\) rate must be multiplied by the hours.

Their reasoning: "It's \(\$60\) per hour plus \(\$120\) for parts, so that's x + 120."

This may lead them to select Choice A (\(f(x) = x + 120\))

Second Most Common Error:

Incomplete INFER reasoning: Students correctly identify the labor cost as 60x but forget about the parts cost entirely, focusing only on the time-dependent portion.

Their reasoning: "The function depends on hours, so it's just 60x for the labor."

This may lead them to select Choice B (\(f(x) = 60x\))

The Bottom Line:

This problem tests whether students can systematically break down a word problem into its mathematical components while keeping track of both variable and fixed costs. The key insight is recognizing that "total cost" means you must account for ALL cost components, not just the time-dependent ones.