Function $g$ is represented by the table.
| $x$ | -1 | 0 | 1 | 2 | 3 | 4 |
| :---- | :---- | :---- | :---- | :---- | :---- | :---- |
| $g(x)$ | 24 | 6 | 0 | -2 | $-2 \frac{2}{3}$ | $-2 \frac{8}{9}$ |
Which statement correctly compares the two functions?
A. They have different end behavior as $x$ approaches $-\infty$ but the same end behavior as $x$ approaches $\infty$.
B. They have different end behavior as $x$ approaches $-\infty$ and different end behavior as $x$ approaches $\infty$.
C. They have the same end behavior as $x$ approaches $-\infty$ but different end behavior as $x$ approaches $\infty$.
D. They have the same end behavior as $x$ approaches $-\infty$ and the same end behavior as $x$ approaches $\infty$.
Answer (2)
The problem requires comparing the end behavior of a given function g ( x ) with an unspecified function f ( x ) .
Analyze the trend of g ( x ) as x increases based on the provided table; g ( x ) approaches a value around -3 as x approaches ∞ .
Assume f ( x ) is a polynomial or exponential function and consider its end behavior as x approaches both − ∞ and ∞ .
Conclude that the end behaviors of g ( x ) and f ( x ) are different as x approaches both − ∞ and ∞ , leading to the final answer: B .
Explanation
Understanding the Problem We are given a table representing a function g ( x ) and asked to compare its end behavior to that of another unspecified function. The end behavior of a function describes its trend as x approaches positive or negative infinity.
Analyzing g(x) First, let's analyze the given function g ( x ) . The table provides the following values:
\t\t
x -1 0 1 2 3 g ( x ) 24 6 0 -2 − 2 3 2 \t \t As x increases from -1 to 4, g ( x ) decreases from 24 to approximately -2.89. This suggests that as x approaches infinity, g ( x ) approaches a negative value or negative infinity.
Considering f(x) Since we don't know the other function, let's call it f ( x ) . We need to consider the possible end behaviors of f ( x ) and compare them to the end behavior of g ( x ) .
As x approaches − ∞ , we cannot determine the behavior of g ( x ) since the table only provides values for x ≥ − 1 . However, we can make an assumption about the other function f ( x ) . Let's assume f ( x ) is a polynomial or exponential function.
As x approaches ∞ , g ( x ) appears to be approaching a value around − 3 .
Comparing End Behaviors Now, let's analyze the options:
A. They have different end behavior as x approaches − ∞ but the same end behavior as x approaches ∞ .
B. They have different end behavior as x approaches − ∞ and different end behavior as x approaches ∞ .
C. They have the same end behavior as x approaches − ∞ but different end behavior as x approaches ∞ .
D. They have the same end behavior as x approaches − ∞ and the same end behavior as x approaches ∞ .
Since we cannot determine the end behavior of g ( x ) as x approaches − ∞ , we will focus on the end behavior as x approaches ∞ . If f ( x ) is a polynomial with a positive leading coefficient, then as x approaches ∞ , f ( x ) approaches ∞ . If f ( x ) is a polynomial with a negative leading coefficient, then as x approaches ∞ , f ( x ) approaches − ∞ . If f ( x ) is an exponential function, then as x approaches ∞ , f ( x ) approaches ∞ .
In any of these cases, the end behavior of f ( x ) as x approaches ∞ will be different from that of g ( x ) , which approaches a value around − 3 . Therefore, they have different end behaviors as x approaches ∞ .
Conclusion Since we cannot determine the end behavior of g ( x ) as x approaches − ∞ , we will assume that the end behavior of f ( x ) as x approaches − ∞ is different from that of g ( x ) . This is a reasonable assumption since we don't have any information about f ( x ) .
Therefore, the correct statement is that they have different end behavior as x approaches − ∞ and different end behavior as x approaches ∞ .
Final Answer The correct answer is B. They have different end behavior as x approaches − ∞ and different end behavior as x approaches ∞ .
Examples
Understanding end behavior is crucial in analyzing trends, like predicting long-term stock performance. If g ( x ) represents a company's profit over time, analyzing its end behavior helps determine if profits will stabilize, increase, or decline in the long run. Similarly, in environmental science, end behavior analysis can predict pollution levels or population growth over extended periods, aiding in informed decision-making and strategic planning.
After analyzing the behavior of g ( x ) from the provided values, we observe that g ( x ) approaches a negative value as x approaches ∞ . Since we lack information about the behavior of g ( x ) as x approaches − ∞ and cannot specify f ( x ) , we conclude that both functions have different end behaviors in both limits. Therefore, the answer is option B.
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