To determine the interval over which the function $f(x)$ is increasing, we need to analyze the values of $f(x)$ corresponding to different intervals of $x$ given in the table. Specifically, we are looking for an interval where, as $x$ increases, $f(x)$ also increases.

Steps:

1. Identify the interval options: We are provided with the following intervals:

   • $(-6, -3)$
   • $(-3, -1)$
   • $(-3, 0)$
   • $(-6, -5)$

2. Check the behavior of $f(x)$ in each interval:

   • For the function to be increasing over an interval, the value of $f(x)$ should increase as $x$ moves from the left endpoint to the right endpoint of the interval.

Let's examine these intervals one by one:

• Interval $(-6, -3)$: Check $f(-6)$ and $f(-3)$. If $f(-3) > f(-6)$, then the function is increasing in this interval.
• Interval $(-3, -1)$: Check $f(-3)$ and $f(-1)$. If $f(-1) > f(-3)$, then the function is increasing in this interval.
• Interval $(-3, 0)$: Check $f(-3)$ and $f(0)$. If $f(0) > f(-3)$, then the function is increasing in this interval.
• Interval $(-6, -5)$: Check $f(-6)$ and $f(-5)$. If $f(-5) > f(-6)$, then the function is increasing in this interval.

Conclusion:

If you have the table of values, you would check the function values for each interval and determine where $f(x)$ increases. However, based on common scenarios:

• If $f(-6) < f(-5)$, the function is increasing in the interval $(-6, -5)$.
• If $f(-6) < f(-3)$, the function is increasing in the interval $(-6, -3)$.

The correct interval for increasing behavior is $(-6, -5)$ if the function value at $x = -5$ is greater than at $x = -6$.

If you want to confirm with specific function values or have any questions, feel free to ask!

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Here are 5 related questions:

1. How can you determine if a function is decreasing using a table?
2. What is the importance of identifying increasing and decreasing intervals in real-life applications?
3. How do critical points relate to the intervals of increasing or decreasing functions?
4. What would happen if the function value remains constant over an interval?
5. How do you graphically determine the intervals where a function increases or decreases?

Tip: Always check the function values at the endpoints of the interval to determine if the function is increasing or decreasing.