Let's analyze the question step-by-step to determine the intervals over which the function [tex]\( f(x) \)[/tex] is positive.
We start with the given table of values for the function [tex]\( f(x) \)[/tex]:
[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]
We need to identify the intervals where [tex]\( f(x) \)[/tex] is positive.
First, let's identify the [tex]\( x \)[/tex] values for which [tex]\( f(x) \)[/tex] is positive:
- For [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 2 \)[/tex] (positive)
- For [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 2 \)[/tex] (positive)
Thus, [tex]\( f(x) \)[/tex] is positive at [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex].
Now, let's analyze each of the provided intervals to see which one correctly describes the interval where [tex]\( f(x) \)[/tex] is positive:
1. [tex]\( (-\infty, 1) \)[/tex]
- [tex]\( x = -3 \)[/tex] (not positive, [tex]\( f(x) = -2 \)[/tex])
- [tex]\( x = -2 \)[/tex] (not positive, [tex]\( f(x) = 0 \)[/tex])
- [tex]\( x = -1 \)[/tex] (positive, [tex]\( f(x) = 2 \)[/tex])
- [tex]\( x = 0 \)[/tex] (positive, [tex]\( f(x) = 2 \)[/tex])
- Conclusion: This interval contains points where [tex]\( f(x) \)[/tex] is not positive.
2. [tex]\( (-2, 1) \)[/tex]
- [tex]\( x = -2 \)[/tex] (not positive, [tex]\( f(x) = 0 \)[/tex])
- [tex]\( x = -1 \)[/tex] (positive, [tex]\( f(x) = 2 \)[/tex])
- [tex]\( x = 0 \)[/tex] (positive, [tex]\( f(x) = 2 \)[/tex])
- Conclusion: This interval correctly describes where [tex]\( f(x) \)[/tex] is positive between the given points.
3. [tex]\( (-\infty, 0) \)[/tex]
- [tex]\( x = -3 \)[/tex] (not positive, [tex]\( f(x) = -2 \)[/tex])
- [tex]\( x = -2 \)[/tex] (not positive, [tex]\( f(x) = 0 \)[/tex])
- [tex]\( x = -1 \)[/tex] (positive, [tex]\( f(x) = 2 \)[/tex])
- Conclusion: This interval contains points where [tex]\( f(x) \)[/tex] is not positive.
4. [tex]\( (1, 4) \)[/tex]
- [tex]\( x = 2 \)[/tex] (not positive, [tex]\( f(x) = -8 \)[/tex])
- [tex]\( x = 3 \)[/tex] (not positive, [tex]\( f(x) = -10 \)[/tex])
- [tex]\( x = 4 \)[/tex] (not positive, [tex]\( f(x) = -20 \)[/tex])
- Conclusion: This interval does not contain any positive values of [tex]\( f(x) \)[/tex].
Based on this analysis, the interval that could be the entire interval over which the function [tex]\( f(x) \)[/tex] is positive is [tex]\( (-2, 1) \)[/tex].
