To determine the end behavior of the function [tex]\(f(x)\)[/tex] given the table of values, we need to analyze how [tex]\(f(x)\)[/tex] behaves as [tex]\(x\)[/tex] approaches positive and negative infinity.
Let's examine the given table of values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-4 & 18 \\
\hline
-3 & 9 \\
\hline
-2 & 6 \\
\hline
-1 & 3 \\
\hline
0 & 0 \\
\hline
1 & -3 \\
\hline
2 & -6 \\
\hline
3 & -9 \\
\hline
4 & -18 \\
\hline
\end{array}
\][/tex]
1. Behavior as [tex]\( x \to \infty \)[/tex]:
- As [tex]\( x \)[/tex] increases from [tex]\( 0 \)[/tex] to [tex]\( 4 \)[/tex] (moving towards positive infinity), the function values [tex]\( f(x) \)[/tex] decrease from [tex]\( 0 \)[/tex] to [tex]\( -18 \)[/tex].
- If the trend continues beyond [tex]\( x = 4 \)[/tex], it suggests that as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( f(x) \)[/tex] will continue to decrease towards [tex]\( -\infty \)[/tex].
2. Behavior as [tex]\( x \to -\infty \)[/tex]:
- As [tex]\( x \)[/tex] decreases from [tex]\( 0 \)[/tex] to [tex]\( -4 \)[/tex] (moving towards negative infinity), the function values [tex]\( f(x) \)[/tex] increase from [tex]\( 0 \)[/tex] to [tex]\( 18 \)[/tex].
- If the trend continues beyond [tex]\( x = -4 \)[/tex], it suggests that as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] will continue to increase towards [tex]\( \infty \)[/tex].
Therefore, the end behavior of the function [tex]\( f(x) \)[/tex] is as follows:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
Given the options, the correct statement describing the end behavior of [tex]\(f(x)\)[/tex] is:
As [tex]\( x \rightarrow \infty, f(x) \rightarrow -\infty \)[/tex], and as [tex]\( x \rightarrow -\infty, f(x) \rightarrow \infty \)[/tex].
