Question

given \\(r(x) = 0.5x - 17\\) and \\(t(x) = -7(1.5)^x\\), which of the following accurately describes the end behavior the two functions have in common?

only the left-end behavior is the same.
neither end behavior is the same.
both the left-end and right-end behaviors are the same.
only the right-end behavior is the same.

Explanation:

Analyze the end behavior of \(r(x)\)

Using the Function Key Features knowledge point

$$
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$$

Analyze the end behavior of \(t(x)\)

Analyze the limits of the exponential function \(t(x) = -7(1.5)^x\) as \(x\) approaches infinity and negative infinity:

• As \(x \to -\infty\), the term \((1.5)^x \to 0\). Therefore, \(t(x) \to -7(0) = 0\).
• As \(x \to \infty\), the term \((1.5)^x \to \infty\). Since it is multiplied by \(-7\), \(t(x) \to -\infty\).

Compare the end behaviors

Compare the left-end and right-end limits of both functions:

• Left-end behavior (\(x \to -\infty\)): \(r(x) \to -\infty\) while \(t(x) \to 0\). These are different.
• Right-end behavior (\(x \to \infty\)): \(r(x) \to \infty\) while \(t(x) \to -\infty\). These are different.
• Therefore, neither end behavior is the same.

Answer:

• Only the left-end behavior is the same.
• Neither end behavior is the same. (Correct answer)
• Both the left-end and right-end behaviors are the same.
• Only the right-end behavior is the same.