Answer :
To determine which table shows a function that is decreasing over the interval [tex]\( (-2,0) \)[/tex], we need to analyze the values of [tex]\( f(x) \)[/tex] at [tex]\( x = -2 \)[/tex], [tex]\( x = -1 \)[/tex], and [tex]\( x = 0 \)[/tex] for each table.
### 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]
For [tex]\( x_1 \)[/tex] in this interval:
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = -5 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
In this table, 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. So, this 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]
For [tex]\( x_2 \)[/tex] in this interval:
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = -15 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -5 \)[/tex]
Here, [tex]\( f(x) \)[/tex] increases from [tex]\( -15 \)[/tex] to [tex]\( -5 \)[/tex] over the interval [tex]\( -2 \)[/tex] to [tex]\( 0 \)[/tex]. Therefore, this 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]
For [tex]\( x_3 \)[/tex] in this interval:
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = -10 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -24 \)[/tex]
In this table, from [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex], [tex]\( f(x) \)[/tex] decreases from 0 to -10. From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex], it further decreases from -10 to -24. Hence, the function is consistently decreasing over the entire interval [tex]\( (-2, 0) \)[/tex].
Therefore, the table that shows a function that is decreasing over the interval [tex]\( (-2,0) \)[/tex] is Table 3.
### 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]
For [tex]\( x_1 \)[/tex] in this interval:
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = -5 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
In this table, 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. So, this 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]
For [tex]\( x_2 \)[/tex] in this interval:
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = -15 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -5 \)[/tex]
Here, [tex]\( f(x) \)[/tex] increases from [tex]\( -15 \)[/tex] to [tex]\( -5 \)[/tex] over the interval [tex]\( -2 \)[/tex] to [tex]\( 0 \)[/tex]. Therefore, this 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]
For [tex]\( x_3 \)[/tex] in this interval:
- At [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- At [tex]\( x = -1 \)[/tex], [tex]\( f(x) = -10 \)[/tex]
- At [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -24 \)[/tex]
In this table, from [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex], [tex]\( f(x) \)[/tex] decreases from 0 to -10. From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex], it further decreases from -10 to -24. Hence, the function is consistently decreasing over the entire interval [tex]\( (-2, 0) \)[/tex].
Therefore, the table that shows a function that is decreasing over the interval [tex]\( (-2,0) \)[/tex] is Table 3.
