To determine the interval over which the function [tex]\( f(x) \)[/tex] is positive, we need to carefully analyze the given values in the provided table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-3 & -2 \\
\hline
-2 & 0 \\
\hline
-1 & 2 \\
\hline
0 & 2 \\
\hline
1 & 0 \\
\hline
2 & -8 \\
\hline
3 & -10 \\
\hline
4 & -20 \\
\hline
\end{array}
\][/tex]
From the table, we observe the following [tex]\( f(x) \)[/tex] values for each [tex]\( x \)[/tex]:
- When [tex]\( x = -3 \)[/tex], [tex]\( f(x) = -2 \)[/tex]
- When [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 2 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 2 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( f(x) = -8 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( f(x) = -10 \)[/tex]
- When [tex]\( x = 4 \)[/tex], [tex]\( f(x) = -20 \)[/tex]
We need to identify the range of [tex]\( x \)[/tex] values where [tex]\( f(x) \)[/tex] is greater than zero. From the table, it is clear that [tex]\( f(x) \)[/tex] is positive at the following points:
- When [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 2 \)[/tex] (positive)
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 2 \)[/tex] (positive)
Hence, [tex]\( f(x) \)[/tex] is positive for [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex].
Next, let's identify the smallest and largest [tex]\( x \)[/tex] values from this subset, which are:
- Minimum [tex]\( x \)[/tex] value: [tex]\( -1 \)[/tex]
- Maximum [tex]\( x \)[/tex] value: [tex]\( 0 \)[/tex]
Now, we need to determine which of the provided intervals encompasses these values. We have the following intervals to consider:
1. [tex]\( (-\infty, 1) \)[/tex]
2. [tex]\( (-2, 1) \)[/tex]
3. [tex]\( (-\infty, 0) \)[/tex]
4. [tex]\( (1, 4) \)[/tex]
From our analysis:
- The interval [tex]\( (-\infty, 1) \)[/tex] includes all [tex]\( x \)[/tex] values from [tex]\( -\infty \)[/tex] to [tex]\( 1 \)[/tex]. This interval includes both [tex]\( -1 \)[/tex] and [tex]\( 0 \)[/tex].
- The interval [tex]\( (-2, 1) \)[/tex] includes all [tex]\( x \)[/tex] values from [tex]\( -2 \)[/tex] to [tex]\( 1 \)[/tex]. This interval also includes both [tex]\( -1 \)[/tex] and [tex]\( 0 \)[/tex].
- The interval [tex]\( (-\infty, 0) \)[/tex] includes all [tex]\( x \)[/tex] values from [tex]\( -\infty \)[/tex] to [tex]\( 0 \)[/tex], but does not include the positive point at [tex]\( x = 0 \)[/tex] when considered as a closed interval at the negative endpoint; it's typically assumed open.
- The interval [tex]\( (1, 4) \)[/tex] includes all [tex]\( x \)[/tex] values from [tex]\( 1 \)[/tex] to [tex]\( 4 \)[/tex], which does not include either [tex]\( -1 \)[/tex] or [tex]\( 0 \)[/tex].
Therefore, the most appropriate interval that could represent the entire range over which the function [tex]\( f(x) \)[/tex] is positive is:
[tex]\[
(-\infty, 1)
\][/tex]
Thus, the correct choice from the provided options is:
[tex]\[
(-\infty, 1)
\][/tex]
