To determine the end behavior of the function [tex]\( f(x) \)[/tex] based on the given data points, let's examine the trend and pattern of the values provided.
The table of values is as follows:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-5 & -6 \\
\hline
-4 & -2 \\
\hline
-3 & 0 \\
\hline
-2 & 4 \\
\hline
-1 & 4 \\
\hline
0 & 0 \\
\hline
1 & -2 \\
\hline
2 & -6 \\
\hline
3 & -10 \\
\hline
\end{array}
\][/tex]
Let's analyze the values as [tex]\( x \)[/tex] increases and decreases:
1. As [tex]\( x \)[/tex] increases:
- When [tex]\( x = -5 \)[/tex], [tex]\( f(x) = -6 \)[/tex]
- When [tex]\( x = -4 \)[/tex], [tex]\( f(x) = -2 \)[/tex]
- When [tex]\( x = -3 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- When [tex]\( x = -2 \)[/tex], [tex]\( f(x) = 4 \)[/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( f(x) = 4 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( f(x) = 0 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( f(x) = -2 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( f(x) = -6 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( f(x) = -10 \)[/tex]
Observing the values, we notice that as [tex]\( x \)[/tex] increases from 0 to 3, [tex]\( f(x) \)[/tex] keeps decreasing.
2. As [tex]\( x \)[/tex] decreases:
- From the positive side, as [tex]\( x \)[/tex] goes from 0 to -5, we see the function [tex]\( f(x) \)[/tex] initially increases and then starts decreasing again.
The general trend suggests that as [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] tends to decrease. Similarly, as [tex]\( x \)[/tex] decreases beyond certain points, [tex]\( f(x) \)[/tex] also ends up decreasing.
Based on the patterns we have observed from the table, the best prediction for the end behavior of the function [tex]\( f(x) \)[/tex] is:
- As [tex]\( x \rightarrow \infty, f(x) \rightarrow -\infty \)[/tex]
- As [tex]\( x \rightarrow -\infty, f(x) \rightarrow -\infty \)[/tex]
Thus, the correct option is:
_As [tex]\( x \rightarrow \infty, f(x) \rightarrow -\infty \)[/tex], and as [tex]\( x \rightarrow -\infty, f(x) \rightarrow -\infty \)[/tex]._
