To determine the end behavior of the function [tex]\( f(x) \)[/tex] based on the given table, we need to carefully analyze the values of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] increases and decreases.
We are given the following table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
x & -5 & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\
\hline
f(x) & 14 & 6 & 0 & -4 & -6 & -6 & -4 & 0 & 6 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Analysis
1. Understanding the Values at the Extremes:
- For [tex]\( x = -5 \)[/tex], [tex]\( f(x) = 14 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( f(x) = 6 \)[/tex]
2. Behavior as [tex]\( x \)[/tex] Increases (Moving from Left to Right in the Table):
- From [tex]\( x = -5 \)[/tex] to [tex]\( x = 0 \)[/tex], [tex]\( f(x) \)[/tex] decreases from 14 to -6.
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 3 \)[/tex], [tex]\( f(x) \)[/tex] increases from -6 to 6.
3. Observing the Pattern:
- After reaching [tex]\( x = 0 \)[/tex], as [tex]\( x \)[/tex] increases further, [tex]\( f(x) \)[/tex] increases from -6 to a positive value (6).
- This indicates a potential trend towards positive values as [tex]\( x \to \infty \)[/tex].
4. Behavior as [tex]\( x \)[/tex] Decreases (Moving from Right to Left in the Table):
- From [tex]\( x = 3 \)[/tex] to [tex]\( x = 0 \)[/tex], [tex]\( f(x) \)[/tex] decreases down to -6.
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = -5 \)[/tex], [tex]\( f(x) \)[/tex] increases back up to 14.
5. Conclusion on End Behavior:
- As [tex]\( x \to \infty \)[/tex], based on the increasing trend from [tex]\( x = 0 \)[/tex] to [tex]\( x = 3 \)[/tex] and the overall pattern, [tex]\( f(x) \)[/tex] appears to move towards positive infinity.
- As [tex]\( x \to - \infty \)[/tex], based on the increasing trend seen when moving back towards negative [tex]\( x \)[/tex]-values, it suggests that [tex]\( f(x) \)[/tex] also tends to positive infinity.
Based on the observed pattern from the table values, we can conclude that:
- As [tex]\( x \rightarrow \infty, f(x) \rightarrow \infty \)[/tex]
- As [tex]\( x \rightarrow - \infty, f(x) \rightarrow \infty \)[/tex]
Therefore, the best statement that describes the end behavior of the graph of [tex]\( f(x) \)[/tex] is:
"As [tex]\( x \rightarrow \infty, f(x) \rightarrow \infty, and as \( x \rightarrow - \infty, f(x) \rightarrow \infty \)[/tex]."
This corresponds to option:
2. [tex]\( \text{As } x \rightarrow \infty, f(x) \rightarrow \infty, \text{ and as } x \rightarrow -\infty, f(x) \rightarrow \infty \)[/tex].
