The function \(\mathrm{f(x) = 200,000(1.21)^x}\) gives a company's predicted annual revenue, in dollars, x years after the company started selling...

1. TRANSLATE the given information

• Given: \(\mathrm{f(x) = 200,000(1.21)^x}\) = predicted annual revenue (in dollars) x years after starting online sales
• Statement to interpret: "\(\mathrm{f(5)}\) is approximately equal to 518,748"
• This means we need to understand what \(\mathrm{f(5)}\) represents in this context

2. INFER what function notation means

• \(\mathrm{f(x)}\) represents the OUTPUT (annual revenue) when the INPUT is x years
• \(\mathrm{f(5)}\) means "what is the annual revenue when \(\mathrm{x = 5}\)?"
• So \(\mathrm{f(5) ≈ 518,748}\) tells us the annual revenue 5 years after starting is about $518,748

3. TRANSLATE this understanding to the answer choices

• We're looking for: "At year 5, the annual revenue is $518,748"
• Choice A matches this interpretation exactly
• Choices B, C, D all discuss relationships or changes, not the actual revenue value

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret what \(\mathrm{f(5)}\) represents, thinking it shows the change in revenue rather than the actual revenue at year 5.

They see "518,748" and think this must be how much revenue increased over 5 years, rather than the total revenue in year 5. This confusion between "value at a point" versus "change over time" is common with function interpretation.

This may lead them to select Choice B (increase by 518,748 dollars).

Second Most Common Error:

Weak INFER skill: Students get distracted by the numbers in the problem context and try to create mathematical relationships that aren't actually being asked for.

Seeing the number 5 in \(\mathrm{f(5)}\), they might think about 5x multipliers or 5% increases, missing that the question is simply asking what \(\mathrm{f(5)}\) means as a function value.

This may lead them to select Choice C (5 times previous year) or Choice D (5% greater).

The Bottom Line:

Function notation problems require clear distinction between "the value of the function at a point" versus "relationships between different points." The key insight is that \(\mathrm{f(5)}\) simply tells us the output when the input is 5, nothing more complex.