To determine the end behavior of the function [tex]\( f(x) \)[/tex] based on the given table, we need to analyze how [tex]\( f(x) \)[/tex] changes as [tex]\( x \)[/tex] increases and decreases.
The table given is:
[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]
First, let's determine the behavior as [tex]\( x \)[/tex] increases (i.e., [tex]\( x \rightarrow \infty \)[/tex]):
- As [tex]\( x \)[/tex] increases from 0 to 4:
- [tex]\( f(0) = 0 \)[/tex]
- [tex]\( f(1) = -3 \)[/tex]
- [tex]\( f(2) = -6 \)[/tex]
- [tex]\( f(3) = -9 \)[/tex]
- [tex]\( f(4) = -18 \)[/tex]
We observe that [tex]\( f(x) \)[/tex] is decreasing as [tex]\( x \)[/tex] increases. Specifically, as [tex]\( x \)[/tex] moves to larger positive values, [tex]\( f(x) \)[/tex] becomes more and more negative. Hence, as [tex]\( x \rightarrow \infty, f(x) \rightarrow -\infty \)[/tex].
Next, let's determine the behavior as [tex]\( x \)[/tex] decreases (i.e., [tex]\( x \rightarrow -\infty \)[/tex]):
- As [tex]\( x \)[/tex] decreases from 0 to -4:
- [tex]\( f(0) = 0 \)[/tex]
- [tex]\( f(-1) = 3 \)[/tex]
- [tex]\( f(-2) = 6 \)[/tex]
- [tex]\( f(-3) = 9 \)[/tex]
- [tex]\( f(-4) = 18 \)[/tex]
We observe that [tex]\( f(x) \)[/tex] is increasing as [tex]\( x \)[/tex] decreases. Specifically, as [tex]\( x \)[/tex] moves to larger negative values, [tex]\( f(x) \)[/tex] becomes more and more positive. Hence, as [tex]\( x \rightarrow -\infty, f(x) \rightarrow \infty \)[/tex].
Therefore, the best prediction for the end behavior of the graph of [tex]\( f(x) \)[/tex] is:
[tex]\[
\text{As } x \rightarrow \infty, f(x) \rightarrow -\infty, \text{ and as } x \rightarrow -\infty, f(x) \rightarrow \infty.
\][/tex]
This corresponds to the third option:
[tex]\[
\text{As } x \rightarrow \infty, f(x) \rightarrow -\infty, \text{ and as } x \rightarrow -\infty, f(x) \rightarrow \infty.
\][/tex]
