Immanuel purchased a certain rare coin on January 1. The function \(\mathrm{f(x) = 65(1.03)}^{x}\), where 0 leq x leq 10,...
GMAT Advanced Math : (Adv_Math) Questions
Immanuel purchased a certain rare coin on January 1. The function \(\mathrm{f(x) = 65(1.03)}^{x}\), where \(\mathrm{0 \leq x \leq 10}\), gives the predicted value, in dollars, of the rare coin x years after Immanuel purchased it. What is the best interpretation of the statement '\(\mathrm{f(8)}\) is approximately equal to \(\mathrm{82}\)' in this context?
When the rare coin's predicted value is approximately \(82\) dollars, it is \(8\%\) greater than the predicted value, in dollars, on January 1 of the previous year.
When the rare coin's predicted value is approximately \(82\) dollars, it is \(8\) times the predicted value, in dollars, on January 1 of the previous year.
From the day Immanuel purchased the rare coin to \(8\) years after Immanuel purchased the coin, its predicted value increased by a total of approximately \(82\) dollars.
\(8\) years after Immanuel purchased the rare coin, its predicted value is approximately \(82\) dollars.
1. TRANSLATE the given information
- Given function: \(\mathrm{f(x) = 65(1.03)^x}\) where \(\mathrm{0 \leq x \leq 10}\)
- This function gives the predicted value, in dollars, of the rare coin x years after purchase
- We need to interpret: "\(\mathrm{f(8)}\) is approximately equal to 82"
2. INFER what \(\mathrm{f(8)}\) represents
- Since \(\mathrm{f(x)}\) gives the predicted value x years after purchase
- \(\mathrm{f(8)}\) must give the predicted value 8 years after purchase
- The input \(\mathrm{x = 8}\) means "8 years after purchase"
- The output \(\mathrm{f(8)}\) means "the predicted value at that time"
3. TRANSLATE the complete statement
- "\(\mathrm{f(8)}\) is approximately equal to 82" means:
- "The predicted value 8 years after purchase is approximately 82 dollars"
- This directly matches answer choice D
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret what \(\mathrm{f(8)}\) represents and focus on the number 8 in the wrong way.
They see "\(\mathrm{f(8) = 82}\)" and think the 8 must relate to "8%" or "8 times" because those concepts appear in choices A and B. They don't properly translate that \(\mathrm{f(8)}\) simply means "the function value when \(\mathrm{x = 8}\)," which represents the predicted value 8 years after purchase.
This may lead them to select Choice A (8% greater) or Choice B (8 times greater).
Second Most Common Error:
Weak INFER skill: Students misunderstand what the function output represents in context.
They correctly identify that \(\mathrm{f(8)}\) involves 8 years, but they think \(\mathrm{f(8) = 82}\) means the coin increased BY 82 dollars over those 8 years, rather than understanding that 82 dollars IS the predicted value at that time.
This may lead them to select Choice C (increased by 82 dollars total).
The Bottom Line:
This problem tests whether students can properly interpret function notation in a real-world context. The key insight is that \(\mathrm{f(8) = 82}\) is simply telling us the output value when the input is 8 - nothing more complex than that.
When the rare coin's predicted value is approximately \(82\) dollars, it is \(8\%\) greater than the predicted value, in dollars, on January 1 of the previous year.
When the rare coin's predicted value is approximately \(82\) dollars, it is \(8\) times the predicted value, in dollars, on January 1 of the previous year.
From the day Immanuel purchased the rare coin to \(8\) years after Immanuel purchased the coin, its predicted value increased by a total of approximately \(82\) dollars.
\(8\) years after Immanuel purchased the rare coin, its predicted value is approximately \(82\) dollars.