Question

x | f(x)
-8 | -16
-6 | -8
-4 | 0
-2 | 8
0 | 16
2 | 32
4 | 64
6 | 128
which could be the entire interval over which the function, f(x), is negative?
○ (-8, -2)
○ (-8, 0)
○ (-∞, -6)
○ (-∞, -4)

Explanation:

Step1: Analyze function values

Check \( f(x) \) values: at \( x = -8 \), \( f(-8) = -16 \) (negative); \( x = -6 \), \( f(-6) = -8 \) (negative); \( x = -4 \), \( f(-4) = 0 \) (not negative); \( x = -2 \), \( f(-2) = 8 \) (positive).

Step2: Evaluate intervals

- Interval \((-8, -2)\): \( x \) between -8 and -2. At \( x = -6 \) (in interval), \( f(x) \) is negative; at \( x = -4 \) (in interval), \( f(x) = 0 \)? Wait, no—wait, \( x = -4 \) is included? No, interval is open. Wait, \( x = -4 \) is at the boundary. Wait, actually, from \( x = -8 \) to \( x = -4 \), \( f(x) \) is negative (since at \( x = -8, -6 \) negative, \( x = -4 \) is 0). But the interval \((-\infty, -4)\): let's check. Wait, the last option (assuming the last is \((-\infty, -4)\)): at \( x < -4 \), like \( x = -8, -6 \), \( f(x) \) is negative. At \( x = -4 \), \( f(x) = 0 \), so for \( x < -4 \), \( f(x) \) is negative? Wait, no—wait the table: \( x = -4 \) is 0, \( x = -2 \) is 8 (positive). Wait, maybe the last option is \((-\infty, -4)\). Wait, let's re - check the options. The options are:
1. \((-8, -2)\): At \( x = -4 \) (in this interval), \( f(-4)=0 \), so \( f(x) \) is not negative here (since 0 is not negative). So this interval has a point where \( f(x)=0 \), so not entirely negative.
2. \((-8, 0)\): At \( x = -2 \) (in interval), \( f(-2)=8 \) (positive), so not entirely negative.
3. \((-\infty, -6)\): Let's see, but we only have data up to \( x=-8 \). But the table shows for \( x < -4 \), like \( x=-8, -6 \), \( f(x) \) is negative. Wait, the fourth option (probably \((-\infty, -4)\)): For \( x < -4 \), \( f(x) \) is negative (since at \( x=-8, -6 \) negative, \( x = -4 \) is 0). Wait, maybe the last option is \((-\infty, -4)\). Wait, but let's check the first option again. Wait, maybe I misread. Wait, the first option is \((-8, -2)\): \( x \) from -8 to -2. At \( x=-8 \), \( f(-8)=-16 \) (negative), \( x=-6 \) (negative), \( x=-4 \) (0, but \( x=-4 \) is in \((-8, -2)\)? Yes, -4 is between -8 and -2. But \( f(-4) = 0 \), so \( f(x) \) is not negative at \( x=-4 \), so this interval is out. The interval \((-\infty, -4)\): For all \( x < -4 \), from the table, \( x=-8, -6 \) (which are \( < -4 \)) have \( f(x) \) negative, and as \( x \) approaches -4 from the left, \( f(x) \) is negative (since at \( x=-4 \), \( f(x)=0 \)). So the interval where \( f(x) \) is entirely negative is \((-\infty, -4)\). Wait, but maybe the last option is \((-\infty, -4)\). Let's confirm with the table:

For \( x < -4 \) (i.e., \( x\in(-\infty, -4) \)):
- When \( x=-8 \), \( f(-8)=-16<0 \)
- When \( x=-6 \), \( f(-6)=-8<0 \)
- As \( x \) approaches -4 from the left, \( f(x) \) is negative (since at \( x = -4 \), \( f(x)=0 \)).

For the other intervals:
- \((-8, -2)\): Contains \( x=-4 \) where \( f(x)=0 \), so not all \( f(x) \) negative.
- \((-8, 0)\): Contains \( x=-2 \) where \( f(x)=8>0 \), so not all \( f(x) \) negative.
- \((-\infty, -6)\): We don't have data for \( x < -8 \), but from the given data, \( x=-8 \) is negative, but we can't be sure if for all \( x < -6 \) (like \( x=-10 \)) \( f(x) \) is negative. But the interval \((-\infty, -4)\) is better because we know for \( x=-8, -6 \) (which are in \((-\infty, -4)\)) \( f(x) \) is negative, and at \( x=-4 \), \( f(x)=0 \), so for \( x < -4 \), \( f(x) \) is negative. Wait, maybe the last option is \((-\infty, -4)\). Assuming the last option is \((-\infty, -4)\), that's the interval where \( f(x) \) is entirely negative.

Wait, maybe I made a mistake earlier. Let's re - examine the options. If the options are:

A. \((-8, -2)\)

B. \((-8, 0)\)

C. \((-\infty, -6)\)

D. \((-\infty, -4)\)

Then:

- Option A: At \( x=-4 \in (-8, -2) \), \( f(-4) = 0 \), so \( f(x) \) is not negative here. Eliminate A.

- Option B: At \( x=-2 \in (-8, 0) \), \( f(-2)=8>0 \), eliminate B.

- Option C: We don't know about \( x < -8 \), but from the table, \( x=-8 \) is negative, but we can't confirm for all \( x < -6 \).

- Option D: For \( x < -4 \), from the table, \( x=-8, -6 \) (which are \( < -4 \)) have \( f(x) \) negative, and at \( x=-4 \), \( f(x)=0 \), so for \( x < -4 \), \( f(x) \) is negative (since when \( x < -4 \), \( x=-8, -6 \) give negative \( f(x) \), and as \( x \) approaches -4 from the left, \( f(x) \) is negative). So the interval where \( f(x) \) is entirely negative is \((-\infty, -4)\).

Answer:

\(\boldsymbol{(-\infty, -4)}\) (assuming the last option is \((-\infty, -4)\); if the options were labeled, e.g., D. \((-\infty, -4)\), then the answer would be D. \((-\infty, -4)\))