To determine which table shows a function that is decreasing over the interval [tex]\((-2, 0)\)[/tex], we need to analyze each table and check the function values within the specified interval. Let's go through each table one by one.
### Table 1:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & 0 \\
\hline
-1 & -5 \\
\hline
0 & 0 \\
\hline
1 & 5 \\
\hline
\end{array}
\][/tex]
In Table 1, we need to look at the values for [tex]\(x = -2\)[/tex], [tex]\(x = -1\)[/tex], and [tex]\(x = 0\)[/tex].
- At [tex]\(x = -2\)[/tex], [tex]\(f(-2) = 0\)[/tex].
- At [tex]\(x = -1\)[/tex], [tex]\(f(-1) = -5\)[/tex].
- At [tex]\(x = 0\)[/tex], [tex]\(f(0) = 0\)[/tex].
We see that from [tex]\(x = -2\)[/tex] to [tex]\(x = -1\)[/tex], [tex]\(f(x)\)[/tex] decreases from 0 to -5. However, from [tex]\(x = -1\)[/tex] to [tex]\(x = 0\)[/tex], [tex]\(f(x)\)[/tex] increases from -5 to 0. Therefore, the function is not consistently decreasing over the entire interval [tex]\((-2, 0)\)[/tex].
### Table 2:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & -15 \\
\hline
0 & -5 \\
\hline
2 & -20 \\
\hline
4 & -30 \\
\hline
\end{array}
\][/tex]
In Table 2, we need to look at the values for [tex]\(x = -2\)[/tex] and [tex]\(x = 0\)[/tex].
- At [tex]\(x = -2\)[/tex], [tex]\(f(-2) = -15\)[/tex].
- At [tex]\(x = 0\)[/tex], [tex]\(f(0) = -5\)[/tex].
We see that the function value increases from -15 to -5 as [tex]\(x\)[/tex] goes from -2 to 0. Thus, the function is not decreasing over the interval [tex]\((-2, 0)\)[/tex].
### Table 3:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-3 & 2 \\
\hline
-2 & 0 \\
\hline
-1 & -10 \\
\hline
0 & -24 \\
\hline
\end{array}
\][/tex]
In Table 3, we need to look at the values for [tex]\(x = -2\)[/tex], [tex]\(x = -1\)[/tex], and [tex]\(x = 0\)[/tex].
- At [tex]\(x = -2\)[/tex], [tex]\(f(-2) = 0\)[/tex].
- At [tex]\(x = -1\)[/tex], [tex]\(f(-1) = -10\)[/tex].
- At [tex]\(x = 0\)[/tex], [tex]\(f(0) = -24\)[/tex].
We see that from [tex]\(x = -2\)[/tex] to [tex]\(x = -1\)[/tex], [tex]\(f(x)\)[/tex] decreases from 0 to -10 and then from [tex]\(x = -1\)[/tex] to [tex]\(x = 0\)[/tex], [tex]\(f(x)\)[/tex] further decreases from -10 to -24. Therefore, the function is consistently decreasing over the interval [tex]\((-2, 0)\)[/tex].
By this detailed analysis, Table 3 is the one that shows a function that is decreasing over the interval [tex]\((-2, 0)\)[/tex].
