To determine the interval where the function [tex]\( k \)[/tex] is increasing, we need to examine the values of [tex]\( k(x) \)[/tex] as [tex]\( x \)[/tex] changes. Here's a detailed analysis based on the table provided:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -1 & 0 & 1 & 2 & 3 & 4 \\
\hline
k(x) & 5 & 8 & 9 & 8 & 5 & 0 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Analysis:
1. Identify Changes in [tex]\( k(x) \)[/tex]:
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]:
- [tex]\( k(-1) = 5 \)[/tex] and [tex]\( k(0) = 8 \)[/tex]. Since [tex]\( 8 > 5 \)[/tex], [tex]\( k(x) \)[/tex] is increasing in this interval.
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]:
- [tex]\( k(0) = 8 \)[/tex] and [tex]\( k(1) = 9 \)[/tex]. Since [tex]\( 9 > 8 \)[/tex], [tex]\( k(x) \)[/tex] is increasing in this interval.
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
- [tex]\( k(1) = 9 \)[/tex] and [tex]\( k(2) = 8 \)[/tex]. Since [tex]\( 9 > 8 \)[/tex], [tex]\( k(x) \)[/tex] is decreasing in this interval.
- From [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
- [tex]\( k(2) = 8 \)[/tex] and [tex]\( k(3) = 5 \)[/tex]. Since [tex]\( 8 > 5 \)[/tex], [tex]\( k(x) \)[/tex] is decreasing in this interval.
- From [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]:
- [tex]\( k(3) = 5 \)[/tex] and [tex]\( k(4) = 0 \)[/tex]. Since [tex]\( 5 > 0 \)[/tex], [tex]\( k(x) \)[/tex] is decreasing in this interval.
2. Determine Intervals:
- We see that the function [tex]\( k \)[/tex] is increasing for [tex]\( x = -1 \)[/tex] to [tex]\( x = 1 \)[/tex].
### Conclusion:
The interval over which the function [tex]\( k \)[/tex] is increasing is from [tex]\( x = -1 \)[/tex] to [tex]\( x = 1 \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{(-1, 1)}
\][/tex]
