To determine the interval where the function is increasing, we need to examine the given table of values for the function [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-3 & 18 \\
\hline
-2 & 3 \\
\hline
-1 & 0 \\
\hline
0 & 3 \\
\hline
1 & 6 \\
\hline
2 & 3 \\
\hline
\end{array}
\][/tex]
First, let's list the points and observe the changes in the function values as [tex]\( x \)[/tex] increases:
- From [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex], [tex]\( f(x) \)[/tex] decreases from 18 to 3.
- From [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex], [tex]\( f(x) \)[/tex] decreases from 3 to 0.
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex], [tex]\( f(x) \)[/tex] increases from 0 to 3.
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex], [tex]\( f(x) \)[/tex] increases from 3 to 6.
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex], [tex]\( f(x) \)[/tex] decreases from 6 to 3.
Next, we identify the intervals where [tex]\( f(x) \)[/tex] is increasing. From the observations above:
- [tex]\( f(x) \)[/tex] is increasing from [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex].
- [tex]\( f(x) \)[/tex] is increasing from [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex].
Thus, the intervals where [tex]\( f(x) \)[/tex] is increasing are [tex]\((-1, 0)\)[/tex] and [tex]\((0, 1)\)[/tex].
Since these two intervals are adjacent and can be combined into a single continuous interval, the largest interval where the function is increasing is:
[tex]\[
(-1, 1)
\][/tex]
Therefore, the largest interval of [tex]\( x \)[/tex] values where the function is increasing is [tex]\((-1, 1)\)[/tex].
