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Advanced Math / Nonlinear functions Difficulty: Hard

$f left parenthesis x right parenthesis equals 9,000 left parenthesis 0.66 right parenthesis Superscript x$

The given function $f$ models the number of advertisements a company sent to its clients each year, where $x$ represents the number of years since $1997$ , and $0 \leq x \leq 5$ . If $y = f (x)$ is graphed in the xy-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?

The minimum estimated number of advertisements the company sent to its clients during the $5$ years was $1,708$ .

The minimum estimated number of advertisements the company sent to its clients during the $5$ years was $9,000$ .

The estimated number of advertisements the company sent to its clients in $1997$ was $1,708$ .

The estimated number of advertisements the company sent to its clients in $1997$ was $9,000$ .

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Explanation

Choice D is correct. The y-intercept of a graph in the xy-plane is the point where $x equals 0$ . For the given function $f$ , the y-intercept of the graph of $y = f (x)$ in the xy-plane can be found by substituting $0$ for $x$ in the equation $y equals 9,000 left parenthesis 0.66 right parenthesis Superscript x$ , which gives $y equals 9,000 left parenthesis 0.66 right parenthesis Superscript 0$ . This is equivalent to $y equals 9,000 left parenthesis 1 right parenthesis$ , or $y equals 9,000$ . Therefore, the y-intercept of the graph of $y = f (x)$ is $left parenthesis 0 comma 9,000 right parenthesis$ . It’s given that the function $f$ models the number of advertisements a company sent to its clients each year. Therefore, $f (x)$ represents the estimated number of advertisements the company sent to its clients each year. It's also given that $x$ represents the number of years since $1997$ . Therefore, $x equals 0$ represents $0$ years since $1997$ , or $1997$ . Thus, the best interpretation of the y-intercept of the graph of $y = f (x)$ is that the estimated number of advertisements the company sent to its clients in $1997$ was $9,000$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.