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Analyzing an Interval

Which table shows a function that is decreasing only over the interval [tex](-1, 1)[/tex]?

[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
-2 & 0 \\
\hline
-1 & 3 \\
\hline
0 & 0 \\
\hline
1 & -3 \\
\hline
2 & 0 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
-2 & 10 \\
\hline
-1 & 8 \\
\hline
0 & 0 \\
\hline
1 & -8 \\
\hline
2 & -10 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
-2 & 0 \\
\hline
-1 & -3 \\
\hline
0 & 0 \\
\hline
1 & 3 \\
\hline
2 & 0 \\
\hline
\end{tabular}
\][/tex]

[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
-2 & -10 \\
\hline
-1 & -8 \\
\hline
0 & 0 \\
\hline
1 & 8 \\
\hline
2 & 10 \\
\hline
\end{tabular}
\][/tex]

Sagot :

GinnyAnsweridn
To determine which table shows a function that is decreasing only over the interval [tex]\((-1, 1)\)[/tex], we'll analyze each table row by row. We need to focus on the values of [tex]\(f(x)\)[/tex] within the interval [tex]\(-1 < x < 1\)[/tex], specifically looking at how these values change.

### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 0 \\ -1 & 3 \\ 0 & 0 \\ 1 & -3 \\ 2 & 0 \\ \hline \end{array} \][/tex]

For [tex]\(x\)[/tex] in [tex]\((-1, 1)\)[/tex], the function's values are:
- [tex]\(f(-1)=3\)[/tex]
- [tex]\(f(0)=0\)[/tex]
- [tex]\(f(1)=-3\)[/tex]

Here, [tex]\(3 \rightarrow 0 \rightarrow -3\)[/tex] which is decreasing.

### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 10 \\ -1 & 8 \\ 0 & 0 \\ 1 & -8 \\ 2 & -10 \\ \hline \end{array} \][/tex]

For [tex]\(x\)[/tex] in [tex]\((-1, 1)\)[/tex], the function's values are:
- [tex]\(f(-1)=8\)[/tex]
- [tex]\(f(0)=0\)[/tex]
- [tex]\(f(1)=-8\)[/tex]

Here, [tex]\(8 \rightarrow 0 \rightarrow -8\)[/tex] which is decreasing.

### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 0 \\ -1 & -3 \\ 0 & 0 \\ 1 & 3 \\ 2 & 0 \\ \hline \end{array} \][/tex]

For [tex]\(x\)[/tex] in [tex]\((-1, 1)\)[/tex], the function's values are:
- [tex]\(f(-1)=-3\)[/tex]
- [tex]\(f(0)=0\)[/tex]
- [tex]\(f(1)=3\)[/tex]

Here, [tex]\(-3 \rightarrow 0 \rightarrow 3\)[/tex] which is increasing.

### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & -10 \\ -1 & -8 \\ 0 & 0 \\ 1 & 8 \\ 2 & 10 \\ \hline \end{array} \][/tex]

For [tex]\(x\)[/tex] in [tex]\((-1, 1)\)[/tex], the function's values are:
- [tex]\(f(-1)=-8\)[/tex]
- [tex]\(f(0)=0\)[/tex]
- [tex]\(f(1)=8\)[/tex]

Here, [tex]\(-8 \rightarrow 0 \rightarrow 8\)[/tex] which is increasing.

### Conclusion
By analyzing each table, we conclude that the functions in both Table 1 and Table 2 are decreasing over the interval [tex]\((-1, 1)\)[/tex]. Therefore, the correct tables are:

[tex]\[ \boxed{1 \text{ and } 2} \][/tex]
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