The function \(\mathrm{f(x) = 240,000(1.22)^x}\) gives a company's predicted annual revenue, in dollars, x years after the company started selling...

1. TRANSLATE the function definition

• Given information:
  • \(\mathrm{f(x) = 240,000(1.22)^x}\)
  • \(\mathrm{f(x)}\) = predicted annual revenue in dollars
  • \(\mathrm{x}\) = years after company started selling jewelry online
  • \(\mathrm{f(5) \approx 648,650}\)

• What this tells us: We need to interpret what \(\mathrm{f(5)}\) means using the function definition

2. INFER what f(5) represents

• Since \(\mathrm{f(x)}\) gives the annual revenue and \(\mathrm{x}\) represents years after starting online:
  • \(\mathrm{f(5)}\) means "the annual revenue when \(\mathrm{x = 5}\)"
  • \(\mathrm{x = 5}\) means "5 years after starting online sales"
  • Therefore: \(\mathrm{f(5)}\) = "annual revenue 5 years after starting online sales"

3. TRANSLATE the mathematical statement to English

• \(\mathrm{f(5) \approx 648,650}\) means:
  • "The annual revenue 5 years after starting online sales is approximately 648,650 dollars"

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what \(\mathrm{f(5)}\) represents in the context

Many students see "\(\mathrm{f(5) \approx 648,650}\)" and think the 5 has some special meaning beyond being the input value. They might think:

• The revenue is 5 times something (Choice C)
• There's a 5% increase (Choice D)
• The 648,650 represents a total increase over 5 years (Choice B)

This happens because they don't carefully connect the function notation \(\mathrm{f(5)}\) to "the value of f when \(\mathrm{x = 5}\)" and then to "the annual revenue 5 years after starting online."

This may lead them to select Choice B, C, or D based on their misconception.

The Bottom Line:

Success on this problem requires carefully translating function notation into contextual language. The key insight is that \(\mathrm{f(5)}\) simply means "plug in \(\mathrm{x = 5}\) and see what the function outputs" - which in this context means "the annual revenue 5 years after starting online sales."