Answer :
To determine over which interval the function [tex]\( f(x) \)[/tex] is positive, we need to examine the tabulated values of [tex]\( f(x) \)[/tex] for different [tex]\( x \)[/tex] values.
Given the data:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & -2 \\ \hline -2 & 0 \\ \hline -1 & 2 \\ \hline 0 & 2 \\ \hline 1 & 0 \\ \hline 2 & -8 \\ \hline 3 & -10 \\ \hline 4 & -20 \\ \hline \end{array} \][/tex]
We observe the function values:
- [tex]\( f(-3) = -2 \)[/tex] (negative)
- [tex]\( f(-2) = 0 \)[/tex] (zero)
- [tex]\( f(-1) = 2 \)[/tex] (positive)
- [tex]\( f(0) = 2 \)[/tex] (positive)
- [tex]\( f(1) = 0 \)[/tex] (zero)
- [tex]\( f(2) = -8 \)[/tex] (negative)
- [tex]\( f(3) = -10 \)[/tex] (negative)
- [tex]\( f(4) = -20 \)[/tex] (negative)
The function [tex]\( f(x) \)[/tex] is positive between [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex], inclusive of these values because [tex]\( f(-1) = 2 \)[/tex] and [tex]\( f(0) = 2 \)[/tex].
Now we evaluate over which of the provided intervals the function [tex]\( f(x) \)[/tex] is positive. The intervals given are:
1. [tex]\((-\infty, 1)\)[/tex]
2. [tex]\((-2, 1)\)[/tex]
3. [tex]\((-\infty, 0)\)[/tex]
4. [tex]\((1, 4)\)[/tex]
Analyzing these intervals:
1. [tex]\((-\infty, 1)\)[/tex]: This interval includes all [tex]\( x \)[/tex] values from negative infinity up to, but not including, [tex]\( x=1 \)[/tex]. Within this interval, the function is positive for [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex], which fits well.
2. [tex]\((-2, 1)\)[/tex]: This interval includes [tex]\( x \)[/tex] values from [tex]\(-2\)[/tex] to [tex]\( 1 \)[/tex]. However, [tex]\( f(-2) = 0 \)[/tex] and [tex]\( f(1) = 0 \)[/tex], which are not positive within this interval.
3. [tex]\((-\infty, 0)\)[/tex]: This interval includes all [tex]\( x \)[/tex] values from negative infinity up to, but not including, [tex]\( x=0 \)[/tex]. Within this interval, the function is positive for [tex]\( x = -1 \)[/tex], but [tex]\( x = 0 \)[/tex] is not included here.
4. [tex]\((1, 4)\)[/tex]: This interval includes [tex]\( x \)[/tex] values from [tex]\( 1 \)[/tex] to [tex]\( 4 \)[/tex]. From the table, we see that all values of [tex]\( f(x) \)[/tex] within this interval are negative or zero.
Among these intervals, the entire interval over which the function [tex]\( f(x) \)[/tex] is positive is:
[tex]\[ (-\infty, 1) \][/tex]
Thus, the correct interval is:
[tex]\[ (-\infty, 1) \][/tex]
Given the data:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -3 & -2 \\ \hline -2 & 0 \\ \hline -1 & 2 \\ \hline 0 & 2 \\ \hline 1 & 0 \\ \hline 2 & -8 \\ \hline 3 & -10 \\ \hline 4 & -20 \\ \hline \end{array} \][/tex]
We observe the function values:
- [tex]\( f(-3) = -2 \)[/tex] (negative)
- [tex]\( f(-2) = 0 \)[/tex] (zero)
- [tex]\( f(-1) = 2 \)[/tex] (positive)
- [tex]\( f(0) = 2 \)[/tex] (positive)
- [tex]\( f(1) = 0 \)[/tex] (zero)
- [tex]\( f(2) = -8 \)[/tex] (negative)
- [tex]\( f(3) = -10 \)[/tex] (negative)
- [tex]\( f(4) = -20 \)[/tex] (negative)
The function [tex]\( f(x) \)[/tex] is positive between [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex], inclusive of these values because [tex]\( f(-1) = 2 \)[/tex] and [tex]\( f(0) = 2 \)[/tex].
Now we evaluate over which of the provided intervals the function [tex]\( f(x) \)[/tex] is positive. The intervals given are:
1. [tex]\((-\infty, 1)\)[/tex]
2. [tex]\((-2, 1)\)[/tex]
3. [tex]\((-\infty, 0)\)[/tex]
4. [tex]\((1, 4)\)[/tex]
Analyzing these intervals:
1. [tex]\((-\infty, 1)\)[/tex]: This interval includes all [tex]\( x \)[/tex] values from negative infinity up to, but not including, [tex]\( x=1 \)[/tex]. Within this interval, the function is positive for [tex]\( x = -1 \)[/tex] and [tex]\( x = 0 \)[/tex], which fits well.
2. [tex]\((-2, 1)\)[/tex]: This interval includes [tex]\( x \)[/tex] values from [tex]\(-2\)[/tex] to [tex]\( 1 \)[/tex]. However, [tex]\( f(-2) = 0 \)[/tex] and [tex]\( f(1) = 0 \)[/tex], which are not positive within this interval.
3. [tex]\((-\infty, 0)\)[/tex]: This interval includes all [tex]\( x \)[/tex] values from negative infinity up to, but not including, [tex]\( x=0 \)[/tex]. Within this interval, the function is positive for [tex]\( x = -1 \)[/tex], but [tex]\( x = 0 \)[/tex] is not included here.
4. [tex]\((1, 4)\)[/tex]: This interval includes [tex]\( x \)[/tex] values from [tex]\( 1 \)[/tex] to [tex]\( 4 \)[/tex]. From the table, we see that all values of [tex]\( f(x) \)[/tex] within this interval are negative or zero.
Among these intervals, the entire interval over which the function [tex]\( f(x) \)[/tex] is positive is:
[tex]\[ (-\infty, 1) \][/tex]
Thus, the correct interval is:
[tex]\[ (-\infty, 1) \][/tex]
