Which table shows a function that is decreasing only over the interval (–1, ∞)?

To determine which table shows a function that is decreasing only over the interval \((-1, \infty)\), let's break down the problem. A function is decreasing on an interval if, as the input (x-value) increases, the output (y-value) decreases.

### Steps to identify the correct table:
1. **Examine the intervals**: You need to check the function values before \(x = -1\) and after \(x = -1\).
2. **Decreasing behavior**:
   - Before \(x = -1\), the function can be either increasing or constant.
   - After \(x = -1\), the function should be consistently decreasing.

### Considerations:
- If \(x < -1\) and \(f(x)\) is increasing, it will not affect our interval of interest because we only care about \((-1, \infty)\).
- If \(x > -1\) and \(f(x)\) starts decreasing, then the table could represent the function you are looking for.

### Example of such a table:
- For \(x < -1\), \(f(x)\) could be increasing or constant.
- For \(x \geq -1\), \(f(x)\) should decrease as \(x\) increases.

#### Sample Table:
| \(x\) | \(f(x)\) |
|-------|----------|
| \(-2\) | 5        |
| \(-1\) | 4        |
|  0     | 3        |
|  1     | 2        |
|  2     | 1        |

In this example:
- For \(x < -1\), \(f(x)\) decreases.
- From \(x = -1\) onwards, \(f(x)\) continues to decrease.

**This table shows a function that decreases over the interval \((-1, \infty)\)**.

Do you need more details or have any questions?

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Here are 5 questions related to this concept:
1. What is the difference between increasing and decreasing functions?
2. How can we identify if a function is increasing or decreasing using its derivative?
3. What are some real-world applications of decreasing functions?
4. How can we determine if a function is strictly decreasing or not?
5. What are the characteristics of a function that is decreasing over all intervals?

**Tip:** Always consider the behavior of the function before and after critical points to determine where it increases or decreases.

Learn how to identify functions that decrease over the interval (-1, ∞) using mathematical analysis. This problem explores the behavior of functions before and after critical points and provides a detailed example table illustrating the concept.

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