Answer :
To determine which tables show a function that is increasing over the interval [tex]\((-2,1)\)[/tex] and nowhere else, we analyze each table separately by examining their values and identifying the behavior in the specified interval.
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & -6 \\ \hline -2 & -3 \\ \hline -1 & -1 \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 6 \\ \hline \end{array} \][/tex]
- [tex]\(\mathbf{-2 < x \leq 1}\)[/tex]:
[tex]\[ \begin{aligned} &f(-2) = -3 \\ &f(-1) = -1 \quad (\text{increasing}) \\ &f(0) = 1 \quad (\text{increasing}) \\ &f(1) = 3 \quad (\text{increasing}) \\ \end{aligned} \][/tex]
In the interval [tex]\((-2,1)\)[/tex], the function is increasing. However, since we are supposed to look for a function that increases only in the interval [tex]\((-2,1)\)[/tex] and nowhere else, we see [tex]\(f(1)=3\)[/tex] is part of this increasing trend and not exclusive to the interval [tex]\((-2,1)\)[/tex].
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & -2 \\ \hline -2 & -4 \\ \hline -1 & -1 \\ \hline 0 & 1 \\ \hline 1 & 4 \\ \hline 2 & 3 \\ \hline \end{array} \][/tex]
- [tex]\(\mathbf{-2 < x \leq 1}\)[/tex]:
[tex]\[ \begin{aligned} &f(-2) = -4 \\ &f(-1) = -1 \quad (\text{increasing}) \\ &f(0) = 1 \quad (\text{increasing}) \\ &f(1) = 4 \quad (\text{increasing}) \\ \end{aligned} \][/tex]
In this interval, the function is increasing. However, the values carry this increasing trend beyond the interval [tex]\((-2,1)\)[/tex] as [tex]\(f(1)=4\)[/tex] is included in the increasing trend.
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & -3 \\ \hline -2 & -5 \\ \hline -1 & -7 \\ \hline 0 & -6 \\ \hline 1 & 1 \\ \hline 2 & -1 \\ \hline \end{array} \][/tex]
- [tex]\(\mathbf{-2 < x \leq 1}\)[/tex]:
[tex]\[ \begin{aligned} &f(-2) = -5 \\ &f(-1) = -7 \quad (\text{decreasing}) \\ &f(0) = -6 \quad (\text{increasing}) \\ &f(1) = 1 \quad (\text{increasing}) \\ \end{aligned} \][/tex]
Within interval [tex]\((-2,1)\)[/tex], the pattern indicates non-monotonic behavior, increasing from x = 0 to x = 1.
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & 5 \\ \hline -2 & 7 \\ \hline -1 & 1 \\ \hline 0 & 0 \\ \hline 1 & -4 \\ \hline 2 & -2 \\ \hline \end{array} \][/tex]
- [tex]\(\mathbf{-2 < x \leq 1}\)[/tex]:
[tex]\[ \begin{aligned} &f(-2) = 7 \\ &f(-1) = 1 \quad (\text{decreasing}) \\ &f(0) = 0 \quad (\text{decreasing}) \\ &f(1) = -4 \\ \end{aligned} \][/tex]
The function does not increase within interval [tex]\((-2,1)\)[/tex].
### Conclusion:
By reviewing the properties and trends within the specified interval [tex]\((-2,1)\)[/tex], the tables 1, 2, and 3 all show an increasing trend within the interval [tex]\( (-2, 1) \)[/tex]. Only table 4 does not show this trend.
Thus, the tables that show functions increasing only over the interval [tex]\((-2,1)\)[/tex] and nowhere else are the tables:
[tex]\[ \boxed{1, 2, 3} \][/tex]
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & -6 \\ \hline -2 & -3 \\ \hline -1 & -1 \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & 6 \\ \hline \end{array} \][/tex]
- [tex]\(\mathbf{-2 < x \leq 1}\)[/tex]:
[tex]\[ \begin{aligned} &f(-2) = -3 \\ &f(-1) = -1 \quad (\text{increasing}) \\ &f(0) = 1 \quad (\text{increasing}) \\ &f(1) = 3 \quad (\text{increasing}) \\ \end{aligned} \][/tex]
In the interval [tex]\((-2,1)\)[/tex], the function is increasing. However, since we are supposed to look for a function that increases only in the interval [tex]\((-2,1)\)[/tex] and nowhere else, we see [tex]\(f(1)=3\)[/tex] is part of this increasing trend and not exclusive to the interval [tex]\((-2,1)\)[/tex].
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & -2 \\ \hline -2 & -4 \\ \hline -1 & -1 \\ \hline 0 & 1 \\ \hline 1 & 4 \\ \hline 2 & 3 \\ \hline \end{array} \][/tex]
- [tex]\(\mathbf{-2 < x \leq 1}\)[/tex]:
[tex]\[ \begin{aligned} &f(-2) = -4 \\ &f(-1) = -1 \quad (\text{increasing}) \\ &f(0) = 1 \quad (\text{increasing}) \\ &f(1) = 4 \quad (\text{increasing}) \\ \end{aligned} \][/tex]
In this interval, the function is increasing. However, the values carry this increasing trend beyond the interval [tex]\((-2,1)\)[/tex] as [tex]\(f(1)=4\)[/tex] is included in the increasing trend.
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & -3 \\ \hline -2 & -5 \\ \hline -1 & -7 \\ \hline 0 & -6 \\ \hline 1 & 1 \\ \hline 2 & -1 \\ \hline \end{array} \][/tex]
- [tex]\(\mathbf{-2 < x \leq 1}\)[/tex]:
[tex]\[ \begin{aligned} &f(-2) = -5 \\ &f(-1) = -7 \quad (\text{decreasing}) \\ &f(0) = -6 \quad (\text{increasing}) \\ &f(1) = 1 \quad (\text{increasing}) \\ \end{aligned} \][/tex]
Within interval [tex]\((-2,1)\)[/tex], the pattern indicates non-monotonic behavior, increasing from x = 0 to x = 1.
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & 5 \\ \hline -2 & 7 \\ \hline -1 & 1 \\ \hline 0 & 0 \\ \hline 1 & -4 \\ \hline 2 & -2 \\ \hline \end{array} \][/tex]
- [tex]\(\mathbf{-2 < x \leq 1}\)[/tex]:
[tex]\[ \begin{aligned} &f(-2) = 7 \\ &f(-1) = 1 \quad (\text{decreasing}) \\ &f(0) = 0 \quad (\text{decreasing}) \\ &f(1) = -4 \\ \end{aligned} \][/tex]
The function does not increase within interval [tex]\((-2,1)\)[/tex].
### Conclusion:
By reviewing the properties and trends within the specified interval [tex]\((-2,1)\)[/tex], the tables 1, 2, and 3 all show an increasing trend within the interval [tex]\( (-2, 1) \)[/tex]. Only table 4 does not show this trend.
Thus, the tables that show functions increasing only over the interval [tex]\((-2,1)\)[/tex] and nowhere else are the tables:
[tex]\[ \boxed{1, 2, 3} \][/tex]
