Question

Based on the table, which best predicts the end behavior of the graph of f(x)?

A) As x→−[infinity], f(x)→0, and as x→[infinity], f(x)→−[infinity]
B) As x→−[infinity], f(x)→0, and as x→[infinity], f(x)→0
C) As x→−[infinity], f(x)→−[infinity], and as x→[infinity], f(x)→−[infinity]
D) As x→−[infinity], f(x)→0, and as x→[infinity], f(x)→−[infinity]

Answer 1

Based on the table, the best prediction for the end behavior of the graph of
`\(f(x)\)` is option D: As
`\(x \to -\infty\),`
`\(f(x) \to 0\)`, and as
`\(x \to \infty\),`
`\(f(x) \to -\infty\).`

Step-by-step explanation:

To determine the end behavior of a function, we examine the values as
`\(x\)` approaches positive and negative infinity. Looking at the given options, option D aligns with the observed behavior in the table. As
`\(x\)` approaches negative infinity,
`\(f(x)\)` tends towards 0, and as
`\(x\)` approaches positive infinity,
`\(f(x)\)` tends towards negative infinity. This behavior is consistent with option D, making it the most suitable prediction.

Analyzing the table values is crucial for understanding the pattern and trend of the function. The provided options describe the limit behavior of the function as
`\(x\)` becomes increasingly large in the positive or negative direction. The correct option not only aligns with the observed values but also reflects a coherent understanding of the function's behavior.

Option D represents the expected end behavior, indicating that as
`\(x\)`becomes extremely negative,
`\(f(x)\)` approaches 0, and as
`\(x\)` becomes extremely positive,
`\(f(x)\)` approaches negative infinity. This prediction is valuable for interpreting the behavior of the function in the long run, aiding in understanding its trends and tendencies beyond the given table values.