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To determine which prediction is valid for the continuous function \( f(x) \), based on the given data, we need to look at the values of \( f(x) \) at specific points and analyze the sign of \( f(x) \) over the intervals:

Here's a summary of the \( f(x) \) values provided:
- \( f(-3) = -15 \)
- \( f(-2) = -5 \)
- \( f(-1) = 0 \)
- \( f(0) = 5 \)
- \( f(1) = 0 \)
- \( f(2) = -5 \)

Observations:
1. From \( x = -1 \) to \( x = 0 \), \( f(x) \) is positive as it changes from 0 to 5.
2. From \( x = 1 \) to \( x = 2 \), \( f(x) \) is negative as it changes from 0 to -5.

Let's examine each prediction:
- \( f(x) \leq 0 \) over the interval \( (-\infty, \infty) \): This is false because \( f(0) = 5 \) (positive).
- \( f(x) > 0 \) over the interval \( (-1, \infty) \): This is false because \( f(x) \) takes negative values and zero values after \( x = -1 \).
- \( f(x) \geq 0 \) over the interval \([-1, 1]\): This is false because while \( f(x) \) is non-negative at \( x = -1, 0, 1 \), it includes positive value at \( x = 0 \).
- \( f(x) < 0 \) over the interval \((0, 2)\): This is true. Both \( f(1) \) and \( f(2) \) are non-positive, with \( f(1) = 0 \) and \( f(2) = -5 \).

Thus, the valid prediction about the continuous function \( f(x) \) is:
- **\( f(x) < 0 \) over the interval \( (0, 2) \)**.

Would you like more details on how these conclusions were reached, or do you have any other questions?

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Here are five related questions you could explore:
1. What would be the behavior of \( f(x) \) beyond the given points, such as \( x > 2 \) or \( x < -3 \)?
2. Can you determine the possible type of function \( f(x) \) could be based on the given values?
3. How does the sign of \( f(x) \) change around the zero points at \( x = -1 \) and \( x = 1 \)?
4. How would the graph of \( f(x) \) look based on the values given?
5. What mathematical properties (like continuity or differentiability) can be inferred from the values of \( f(x) \)?

**Tip:** When trying to analyze a function's behavior over an interval, consider plotting the points and connecting them to visualize changes in the function's sign and value.

Which is a valid prediction about the continuous function f(x)? The table provides values of f(x) for different x-values: -3, -2, -1, 0, 1, and 2.

Explore how to determine valid predictions about a continuous function f(x) using data values for x between -3 and 2. This problem examines the behavior of the function on various intervals, focusing on sign changes and continuous behavior. Suitable for high school students learning about function analysis.

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