Answer :
Let's analyze each of the predictions about the function \( f(x) \) using the given data points.
1. Prediction: \( f(x) \geq 0 \) over the interval \([5, \infty)\)
- We need to check if \( f(x) \) is non-negative (i.e., \( f(x) \geq 0 \)) for all \( x \geq 5 \).
- Looking at the data for \( x \geq 5 \):
- When \( x = 5 \), \( f(5) = 0 \).
- When \( x = 7 \), \( f(7) = 4 \).
- Both \( f(5) \) and \( f(7) \) are non-negative.
Therefore, this prediction is valid.
2. Prediction: \( f(x) \leq 0 \) over the interval \([-1, \infty)\)
- We need to check if \( f(x) \) is non-positive (i.e., \( f(x) \leq 0 \)) for all \( x \geq -1 \).
- Looking at the data for \( x \geq -1 \):
- When \( x = -1 \), \( f(-1) = 0 \).
- When \( x = 1 \), \( f(1) = -2 \).
- When \( x = 3 \), \( f(3) = -2 \).
- When \( x = 5 \), \( f(5) = 0 \).
- When \( x = 7 \), \( f(7) = 4 \).
- Here, \( f(7) = 4 \) is not non-positive.
Therefore, this prediction is invalid.
3. Prediction: \( f(x) > 0 \) over the interval \((-\infty, 1)\)
- We need to check if \( f(x) \) is positive (i.e., \( f(x) > 0 \)) for all \( x < 1 \).
- Looking at the data for \( x < 1 \):
- When \( x = -5 \), \( f(-5) = 8 \).
- When \( x = -3 \), \( f(-3) = 4 \).
- When \( x = -1 \), \( f(-1) = 0 \).
- Here, \( f(-1) = 0 \) is not positive.
Therefore, this prediction is invalid.
4. Prediction: \( f(x) < 0 \) over the interval \((-\infty,-1)\)
- We need to check if \( f(x) \) is negative (i.e., \( f(x) < 0 \)) for all \( x < -1 \).
- Looking at the data for \( x < -1 \):
- When \( x = -5 \), \( f(-5) = 8 \).
- When \( x = -3 \), \( f(-3) = 4 \).
Therefore, this prediction is invalid.
In conclusion, the only valid prediction about the continuous function \( f(x) \) is:
[tex]\[ f(x) \geq 0 \, \text{over the interval} \, [5, \infty). \][/tex]
1. Prediction: \( f(x) \geq 0 \) over the interval \([5, \infty)\)
- We need to check if \( f(x) \) is non-negative (i.e., \( f(x) \geq 0 \)) for all \( x \geq 5 \).
- Looking at the data for \( x \geq 5 \):
- When \( x = 5 \), \( f(5) = 0 \).
- When \( x = 7 \), \( f(7) = 4 \).
- Both \( f(5) \) and \( f(7) \) are non-negative.
Therefore, this prediction is valid.
2. Prediction: \( f(x) \leq 0 \) over the interval \([-1, \infty)\)
- We need to check if \( f(x) \) is non-positive (i.e., \( f(x) \leq 0 \)) for all \( x \geq -1 \).
- Looking at the data for \( x \geq -1 \):
- When \( x = -1 \), \( f(-1) = 0 \).
- When \( x = 1 \), \( f(1) = -2 \).
- When \( x = 3 \), \( f(3) = -2 \).
- When \( x = 5 \), \( f(5) = 0 \).
- When \( x = 7 \), \( f(7) = 4 \).
- Here, \( f(7) = 4 \) is not non-positive.
Therefore, this prediction is invalid.
3. Prediction: \( f(x) > 0 \) over the interval \((-\infty, 1)\)
- We need to check if \( f(x) \) is positive (i.e., \( f(x) > 0 \)) for all \( x < 1 \).
- Looking at the data for \( x < 1 \):
- When \( x = -5 \), \( f(-5) = 8 \).
- When \( x = -3 \), \( f(-3) = 4 \).
- When \( x = -1 \), \( f(-1) = 0 \).
- Here, \( f(-1) = 0 \) is not positive.
Therefore, this prediction is invalid.
4. Prediction: \( f(x) < 0 \) over the interval \((-\infty,-1)\)
- We need to check if \( f(x) \) is negative (i.e., \( f(x) < 0 \)) for all \( x < -1 \).
- Looking at the data for \( x < -1 \):
- When \( x = -5 \), \( f(-5) = 8 \).
- When \( x = -3 \), \( f(-3) = 4 \).
Therefore, this prediction is invalid.
In conclusion, the only valid prediction about the continuous function \( f(x) \) is:
[tex]\[ f(x) \geq 0 \, \text{over the interval} \, [5, \infty). \][/tex]
