Answer :
To determine which table shows a function that is decreasing only over the interval [tex]\((-1,1)\)[/tex], we need to analyze each table to see if the values of the function decrease as [tex]\(x\)[/tex] moves from [tex]\(-1\)[/tex] to 0 to 1.
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 0 \\ \hline -1 & 3 \\ \hline 0 & 0 \\ \hline 1 & -3 \\ \hline 2 & 0 \\ \hline \end{array} \][/tex]
This function shows:
- [tex]\(f(-1) = 3\)[/tex]
- [tex]\(f(0) = 0\)[/tex]
- [tex]\(f(1) = -3\)[/tex]
Here, [tex]\(f(x)\)[/tex] is decreasing from [tex]\(3\)[/tex] to [tex]\(0\)[/tex] to [tex]\(-3\)[/tex] over the interval [tex]\((-1,1)\)[/tex].
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 10 \\ \hline -1 & 8 \\ \hline 0 & 0 \\ \hline 1 & -8 \\ \hline 2 & -10 \\ \hline \end{array} \][/tex]
This function shows:
- [tex]\(f(-1) = 8\)[/tex]
- [tex]\(f(0) = 0\)[/tex]
- [tex]\(f(1) = -8\)[/tex]
Here, [tex]\(f(x)\)[/tex] is also decreasing from [tex]\(8\)[/tex] to [tex]\(0\)[/tex] to [tex]\(-8\)[/tex] over the interval [tex]\((-1,1)\)[/tex].
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 0 \\ \hline -1 & -3 \\ \hline 0 & 0 \\ \hline 1 & 3 \\ \hline 2 & 0 \\ \hline \end{array} \][/tex]
This function shows:
- [tex]\(f(-1) = -3\)[/tex]
- [tex]\(f(0) = 0\)[/tex]
- [tex]\(f(1) = 3\)[/tex]
Here, [tex]\(f(x)\)[/tex] is not decreasing over the interval [tex]\((-1,1)\)[/tex], since [tex]\(f(x)\)[/tex] increases from [tex]\(-3\)[/tex] to [tex]\(0\)[/tex] and then to [tex]\(3\)[/tex].
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & -10 \\ \hline -1 & -8 \\ \hline 0 & 0 \\ \hline 1 & 8 \\ \hline 2 & 10 \\ \hline \end{array} \][/tex]
This function shows:
- [tex]\(f(-1) = -8\)[/tex]
- [tex]\(f(0) = 0\)[/tex]
- [tex]\(f(1) = 8\)[/tex]
Here, [tex]\(f(x)\)[/tex] is not decreasing over the interval [tex]\((-1,1)\)[/tex], since [tex]\(f(x)\)[/tex] increases from [tex]\(-8\)[/tex] to [tex]\(0\)[/tex] and then to [tex]\(8\)[/tex].
From the analysis, Table 1 and Table 2 exhibit decreasing behavior over the interval [tex]\((-1,1)\)[/tex]. However, the question specifies that the function should be decreasing only over this interval. Despite both these tables meeting the condition, the required answer seems to be more explicitly Table 1.
Therefore, the table that shows a function that is decreasing only over the interval [tex]\((-1,1)\)[/tex] is:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 0 \\ \hline -1 & 3 \\ \hline 0 & 0 \\ \hline 1 & -3 \\ \hline 2 & 0 \\ \hline \end{array} \][/tex]
Thus, the correct table is:
1
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 0 \\ \hline -1 & 3 \\ \hline 0 & 0 \\ \hline 1 & -3 \\ \hline 2 & 0 \\ \hline \end{array} \][/tex]
This function shows:
- [tex]\(f(-1) = 3\)[/tex]
- [tex]\(f(0) = 0\)[/tex]
- [tex]\(f(1) = -3\)[/tex]
Here, [tex]\(f(x)\)[/tex] is decreasing from [tex]\(3\)[/tex] to [tex]\(0\)[/tex] to [tex]\(-3\)[/tex] over the interval [tex]\((-1,1)\)[/tex].
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 10 \\ \hline -1 & 8 \\ \hline 0 & 0 \\ \hline 1 & -8 \\ \hline 2 & -10 \\ \hline \end{array} \][/tex]
This function shows:
- [tex]\(f(-1) = 8\)[/tex]
- [tex]\(f(0) = 0\)[/tex]
- [tex]\(f(1) = -8\)[/tex]
Here, [tex]\(f(x)\)[/tex] is also decreasing from [tex]\(8\)[/tex] to [tex]\(0\)[/tex] to [tex]\(-8\)[/tex] over the interval [tex]\((-1,1)\)[/tex].
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 0 \\ \hline -1 & -3 \\ \hline 0 & 0 \\ \hline 1 & 3 \\ \hline 2 & 0 \\ \hline \end{array} \][/tex]
This function shows:
- [tex]\(f(-1) = -3\)[/tex]
- [tex]\(f(0) = 0\)[/tex]
- [tex]\(f(1) = 3\)[/tex]
Here, [tex]\(f(x)\)[/tex] is not decreasing over the interval [tex]\((-1,1)\)[/tex], since [tex]\(f(x)\)[/tex] increases from [tex]\(-3\)[/tex] to [tex]\(0\)[/tex] and then to [tex]\(3\)[/tex].
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & -10 \\ \hline -1 & -8 \\ \hline 0 & 0 \\ \hline 1 & 8 \\ \hline 2 & 10 \\ \hline \end{array} \][/tex]
This function shows:
- [tex]\(f(-1) = -8\)[/tex]
- [tex]\(f(0) = 0\)[/tex]
- [tex]\(f(1) = 8\)[/tex]
Here, [tex]\(f(x)\)[/tex] is not decreasing over the interval [tex]\((-1,1)\)[/tex], since [tex]\(f(x)\)[/tex] increases from [tex]\(-8\)[/tex] to [tex]\(0\)[/tex] and then to [tex]\(8\)[/tex].
From the analysis, Table 1 and Table 2 exhibit decreasing behavior over the interval [tex]\((-1,1)\)[/tex]. However, the question specifies that the function should be decreasing only over this interval. Despite both these tables meeting the condition, the required answer seems to be more explicitly Table 1.
Therefore, the table that shows a function that is decreasing only over the interval [tex]\((-1,1)\)[/tex] is:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 0 \\ \hline -1 & 3 \\ \hline 0 & 0 \\ \hline 1 & -3 \\ \hline 2 & 0 \\ \hline \end{array} \][/tex]
Thus, the correct table is:
1
