To determine the intervals where the function [tex]\( f(x) \)[/tex] is increasing, let's carefully analyze the given table by comparing consecutive values of [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-6 & 34 \\
\hline
-5 & 3 \\
\hline
-4 & -10 \\
\hline
-3 & -11 \\
\hline
-2 & -6 \\
\hline
-1 & -1 \\
\hline
0 & -2 \\
\hline
1 & -15 \\
\hline
\end{array}
\][/tex]
Now, let's look at the changes between these [tex]\( f(x) \)[/tex] values for consecutive [tex]\( x \)[/tex] values:
1. From [tex]\( x = -6 \)[/tex] to [tex]\( x = -5 \)[/tex]: [tex]\( f(-6) = 34 \)[/tex] to [tex]\( f(-5) = 3 \)[/tex] (decreasing)
2. From [tex]\( x = -5 \)[/tex] to [tex]\( x = -4 \)[/tex]: [tex]\( f(-5) = 3 \)[/tex] to [tex]\( f(-4) = -10 \)[/tex] (decreasing)
3. From [tex]\( x = -4 \)[/tex] to [tex]\( x = -3 \)[/tex]: [tex]\( f(-4) = -10 \)[/tex] to [tex]\( f(-3) = -11 \)[/tex] (decreasing)
4. From [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]: [tex]\( f(-3) = -11 \)[/tex] to [tex]\( f(-2) = -6 \)[/tex] (increasing)
5. From [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]: [tex]\( f(-2) = -6 \)[/tex] to [tex]\( f(-1) = -1 \)[/tex] (increasing)
6. From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\( f(-1) = -1 \)[/tex] to [tex]\( f(0) = -2 \)[/tex] (decreasing)
7. From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\( f(0) = -2 \)[/tex] to [tex]\( f(1) = -15 \)[/tex] (decreasing)
We observe that the function [tex]\( f(x) \)[/tex] is increasing in two intervals:
- From [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]
- From [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]
Therefore, the correct interval from the given choices where the function [tex]\( f(x) \)[/tex] is increasing is from [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]:
[tex]\[ (-3, -2) \][/tex]
Hence, the correct answer is:
[tex]\[ (-3, -2) \][/tex]
