x | f(x)\n-8 | -16\n-6 | -8\n-4 | 0\n-2 | 8\n0 | 16\n2 | 32\n4 | 64\n6 | 128\nwhich could be the entire…

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)))