The total cost \(\mathrm{f(x)}\), in dollars, to lease a car for 36 months from a particular car dealership is given...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = 36x + 1,000}\) (total cost for 36-month lease)
  • \(\mathrm{x}\) represents monthly payment in dollars
  • Need to find total cost when monthly payment is $400

• What this tells us: We need to find \(\mathrm{f(400)}\)

2. TRANSLATE what we need to do

• "When the monthly payment is $400" means substitute \(\mathrm{x = 400}\) into our function
• We need to calculate \(\mathrm{f(400) = 36(400) + 1,000}\)

3. SIMPLIFY the calculation

• \(\mathrm{f(400) = 36(400) + 1,000}\)
• \(\mathrm{f(400) = 14,400 + 1,000}\)
• \(\mathrm{f(400) = 15,400}\)

Answer: C. $15,400

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making a sign error during calculation

Students might accidentally subtract instead of add: \(\mathrm{f(400) = 36(400) - 1,000 = 14,400 - 1,000 = 13,400}\)

This may lead them to select Choice A ($13,400)

Second Most Common Error:

Poor TRANSLATE reasoning: Confusion about function notation

Students might not understand that \(\mathrm{f(400)}\) means "substitute 400 for \(\mathrm{x}\)" and instead try to manipulate the equation in other ways, leading to calculation errors and random arithmetic mistakes.

This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

This problem tests whether students can properly substitute values into functions and perform basic arithmetic accurately. The key is recognizing that "monthly payment is $400" directly translates to evaluating \(\mathrm{f(400)}\).