Answer :
Let's examine each prediction in detail and verify their validity using the given table.
### Prediction 1: \( f(x) \geq 0 \) over the interval \([5, \infty)\)
We need to check the values of \( f(x) \) for \( x \geq 5 \):
- For \( x = 5 \), \( f(5) = 0 \)
- For \( x = 7 \), \( f(7) = 4 \)
Both of these values are greater than or equal to 0, so this prediction is correct.
### Prediction 2: \( f(x) \leq 0 \) over the interval \([-1, \infty)\)
We need to check the values of \( f(x) \) for \( x \geq -1 \):
- For \( x = -1 \), \( f(-1) = 0 \)
- For \( x = 1 \), \( f(1) = -2 \)
- For \( x = 3 \), \( f(3) = -2 \)
- For \( x = 5 \), \( f(5) = 0 \)
- For \( x = 7 \), \( f(7) = 4 \)
The value \( f(7) = 4 \) is greater than 0, so this prediction is incorrect.
### Prediction 3: \( f(x) > 0 \) over the interval \((-\infty, 1)\)
We need to check the values of \( f(x) \) for \( x < 1 \):
- For \( x = -5 \), \( f(-5) = 8 \)
- For \( x = -3 \), \( f(-3) = 4 \)
- For \( x = -1 \), \( f(-1) = 0 \)
The value \( f(-1) = 0 \) is not greater than 0, so this prediction is incorrect.
### Prediction 4: \( f(x) < 0 \) over the interval \((-\infty, -1)\)
We need to check the values of \( f(x) \) for \( x < -1 \):
- For \( x = -5 \), \( f(-5) = 8 \)
- For \( x = -3 \), \( f(-3) = 4 \)
Both of these values are greater than 0, so this prediction is incorrect.
Based on the verification:
- Prediction 1 is valid.
- Predictions 2, 3, and 4 are invalid.
Thus, the correct and valid prediction about the continuous function \( f(x) \) is:
[tex]\[ f(x) \geq 0 \text{ over the interval } [5, \infty). \][/tex]
### Prediction 1: \( f(x) \geq 0 \) over the interval \([5, \infty)\)
We need to check the values of \( f(x) \) for \( x \geq 5 \):
- For \( x = 5 \), \( f(5) = 0 \)
- For \( x = 7 \), \( f(7) = 4 \)
Both of these values are greater than or equal to 0, so this prediction is correct.
### Prediction 2: \( f(x) \leq 0 \) over the interval \([-1, \infty)\)
We need to check the values of \( f(x) \) for \( x \geq -1 \):
- For \( x = -1 \), \( f(-1) = 0 \)
- For \( x = 1 \), \( f(1) = -2 \)
- For \( x = 3 \), \( f(3) = -2 \)
- For \( x = 5 \), \( f(5) = 0 \)
- For \( x = 7 \), \( f(7) = 4 \)
The value \( f(7) = 4 \) is greater than 0, so this prediction is incorrect.
### Prediction 3: \( f(x) > 0 \) over the interval \((-\infty, 1)\)
We need to check the values of \( f(x) \) for \( x < 1 \):
- For \( x = -5 \), \( f(-5) = 8 \)
- For \( x = -3 \), \( f(-3) = 4 \)
- For \( x = -1 \), \( f(-1) = 0 \)
The value \( f(-1) = 0 \) is not greater than 0, so this prediction is incorrect.
### Prediction 4: \( f(x) < 0 \) over the interval \((-\infty, -1)\)
We need to check the values of \( f(x) \) for \( x < -1 \):
- For \( x = -5 \), \( f(-5) = 8 \)
- For \( x = -3 \), \( f(-3) = 4 \)
Both of these values are greater than 0, so this prediction is incorrect.
Based on the verification:
- Prediction 1 is valid.
- Predictions 2, 3, and 4 are invalid.
Thus, the correct and valid prediction about the continuous function \( f(x) \) is:
[tex]\[ f(x) \geq 0 \text{ over the interval } [5, \infty). \][/tex]
