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?
option 1: (-8, -2)
option 2: (-8, 0)
option 3: (-∞, -6)
option 4: (-∞, -4)

Explanation:

Step1: Analyze function values

From the table, when \( x = -8 \), \( f(x) = -16 \) (negative); \( x = -6 \), \( f(x) = -8 \) (negative); \( x = -4 \), \( f(x) = 0 \); \( x = -2 \), \( f(x) = 8 \) (positive). So we need an interval where all \( f(x) \) are negative.

Step2: Evaluate each option

• Option 1: \((-8, -2)\): At \( x=-4 \) (in this interval), \( f(x)=0 \) (not negative), so invalid.
• Option 2: \((-8, 0)\): At \( x=-2 \) (in this interval), \( f(x)=8 \) (positive), invalid.
• Option 3: \((-\infty, -6)\): At \( x=-8 \) (in this interval), \( f(x)=-16 \) (negative), but \( x=-6 \) is not included, but let's check next. Wait, \( x=-6 \) has \( f(x)=-8 \) (negative), but the interval is \((-\infty, -6)\), so \( x=-6 \) is not in it. Wait, no, let's re - check. Wait, the function at \( x=-8 \) is - 16, \( x=-6 \) is - 8, \( x=-4 \) is 0. Wait, the interval \((-\infty, -4)\): For \( x < - 4\) (like \( x=-8, -6\)), \( f(x) \) is negative ( - 16, - 8), and at \( x=-4 \), \( f(x)=0 \). So \((-\infty, -4)\) is an interval where all \( x \) in it have \( f(x)<0 \) (since at \( x=-4 \), \( f(x)=0 \), not included). Let's check the third option \((-\infty, -6)\): If we take \( x=-7 \) (in \((-\infty, -6)\)), \( f(x) \) would be between - 16 and - 8 (negative), but the interval \((-\infty, -4)\) includes more values where \( f(x) \) is negative. Wait, let's re - evaluate each option:
• For \((-\infty, -4)\): All \( x < - 4\) (e.g., \( x=-8, -6\)) have \( f(x)<0 \), and at \( x = - 4\), \( f(x)=0 \) (not included in the open interval).
• For \((-\infty, -6)\): It's a subset of \((-\infty, -4)\), but we need the entire interval where \( f(x) \) is negative. Since at \( x=-6 \), \( f(x)=-8 \) (negative), but the interval \((-\infty, -4)\) includes \( x=-6, -8\) etc., and at \( x=-4 \), \( f(x)=0 \) (not in the interval). So \((-\infty, -4)\) is the interval where for all \( x \) in the interval, \( f(x)<0 \).

Answer:

\((-\infty, -4)\) (the option corresponding to \((-\infty, -4)\))