Question

describing a relationship with a function

the table shows the value of a car as it relates to its age.

which statements describe the value of the car as a function of \\(x\\), its age in years? check all that apply.

• the relationship is quadratic.
• the relationship is exponential.
• the domain is \\(\\{x \mid x < 0\\}\\).
• the range is \\(\\{v \mid 0 \le v \le 20,000\\}\\).
• the value, \\(v\\), is represented by the equation \\(v = 20,000(0.9)^x\\).

Explanation:

Analyze the ratio of successive values

Using the Exponential Growth and Decay Behavior knowledge point

$$
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$$

Since the ratio of successive values is constant at \(0.9\), the relationship is exponential, represented by \(V = 20,000(0.9)^x\).

Determine the domain of the function

Using the Domain of Exponential Functions knowledge point

$$ x \ge 0 $$

The age of a car, \(x\), represents time in years since purchase, which cannot be negative. Thus, the domain is \(\{x \mid x \ge 0\}\). The statement "The domain is \(\{x \mid x < 0\}\)" is incorrect.

Determine the range of the function

Using the Range of Exponential Functions knowledge point

$$ 0 \le V \le 20,000 $$

The initial value of the car is \(\$20,000\), and as time \(x \ge 0\) increases, the value decreases exponentially but remains non-negative. Thus, the range of the function in this context is \(\{V \mid 0 \le V \le 20,000\}\).

Answer:

• The relationship is quadratic.
• The relationship is exponential. (Correct answer)
• The domain is {x | x &lt; 0}
• The range is {V | 0 &le; V &le; 20,000} (Correct answer)
• The value, V, is represented by the equation V = 20,000(0.9)^x (Correct answer)