To determine where the function [tex]\( f(x) \)[/tex] is increasing, we need to examine the given [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] values.
Here is the table we have:
[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]
We look for intervals where [tex]\( f(x) \)[/tex] increases as [tex]\( x \)[/tex] increases.
1. From [tex]\( x = -6 \)[/tex] to [tex]\( x = -5 \)[/tex]; [tex]\( f(-6) = 34 \)[/tex] and [tex]\( f(-5) = 3 \)[/tex]
- [tex]\( f(-6) > f(-5) \)[/tex] (decreasing)
2. From [tex]\( x = -5 \)[/tex] to [tex]\( x = -4 \)[/tex]; [tex]\( f(-5) = 3 \)[/tex] and [tex]\( f(-4) = -10 \)[/tex]
- [tex]\( f(-5) > f(-4) \)[/tex] (decreasing)
3. From [tex]\( x = -4 \)[/tex] to [tex]\( x = -3 \)[/tex]; [tex]\( f(-4) = -10 \)[/tex] and [tex]\( f(-3) = -11 \)[/tex]
- [tex]\( f(-4) > f(-3) \)[/tex] (decreasing)
4. From [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]; [tex]\( f(-3) = -11 \)[/tex] and [tex]\( f(-2) = -6 \)[/tex]
- [tex]\( f(-3) < f(-2) \)[/tex] (increasing)
5. From [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]; [tex]\( f(-2) = -6 \)[/tex] and [tex]\( f(-1) = -1 \)[/tex]
- [tex]\( f(-2) < f(-1) \)[/tex] (increasing)
6. From [tex]\( x = 0 \)[/tex]; [tex]\( f(0) = -2 \)[/tex]
- [tex]\( x = -1 \)[/tex] to [tex]\( x = 0; f(-1) = -1 \begins an increase and \(f(0)\ decreases decreasing
7. From \( x = 1) ; f(0) = -1; and \( f(1 = -15, 0) \ commences an increase and
oint
Checking each \( x \)[/tex]:
- The interval from \( x(-3, -2) \( -3 and \(-2( increasing becauseif (\( f(-3)\ <\ f(-2) )
- The interval \( x(-2, -1) \( from becauseif \( f(-2 incrementing -1)
Thus, the function \( f(x) is only between \( -3 -2 and \( -2 -1 respectively.
So, the correct interval where the function is increasing is:
\(\boxed{-3, -1}).
