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Advanced Math / Nonlinear functions Difficulty: Medium

Immanuel purchased a certain rare coin on January 1. The function $f left parenthesis x right parenthesis equals 65 left parenthesis 1.03 right parenthesis Superscript x$ , where $0 \leq x \leq 10$ , gives the predicted value, in dollars, of the rare coin $x$ years after Immanuel purchased it. What is the best interpretation of the statement “ $f left parenthesis 8 right parenthesis$ is approximately equal to $82$ ” in this context?

When the rare coin's predicted value is approximately $82$ dollars, it is $8 percent sign$ 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.

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Explanation

Choice D is correct. It’s given that the function $f left parenthesis x right parenthesis equals 65 left parenthesis 1.03 right parenthesis Superscript x$ gives the predicted value, in dollars, of a certain rare coin $x$ years after Immanuel purchased it. It follows that $f (x)$ represents the predicted value, in dollars, of the coin $x$ years after Immanuel purchased it. Since the value of $f left parenthesis 8 right parenthesis$ is the value of $f (x)$ when $x equals 8$ , it follows that “ $f left parenthesis 8 right parenthesis$ is approximately equal to $82$ ” means that $f (x)$ is approximately equal to $82$ when $x equals 8$ . Therefore, the best interpretation of the statement “ $f left parenthesis 8 right parenthesis$ is approximately equal to $82$ ” in this context is $8$ years after Immanuel purchased the rare coin, its predicted value is approximately $82$ dollars.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.