Answer :
To determine the end behavior of the function \( f(x) \), we need to analyze the trends in the values of \( x \) and \( f(x) \) given in the table. The table provides the following values:
[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]
From the table, we observe the following:
1. As \( x \) increases from \(-5\) to \(3\), the function \( f(x) \) starts at \( 14 \), decreases to \( -6 \) around \( x = -1 \), and then increases back to \( 6 \) as \( x \) reaches \( 3 \).
2. This pattern suggests that \( f(x) \) tends to increase again as \( x \) continues to grow positively.
3. Similarly, If we consider \( x \) becoming more negative from \(-5\) onwards, \( f(x) \) is seen to start high at \( 14 \), decrease significantly, and shows a potential to start increasing beyond the given negative values.
Based on these observations, the trend indicates:
- As \( x \rightarrow \infty \), \( f(x) \rightarrow \infty \)
- As \( x \rightarrow -\infty \), \( f(x) \rightarrow \infty \)
So, the statement that best describes the prediction for the end behavior of the graph of \( f(x) \) is:
"As \( x \rightarrow \infty, f(x) \rightarrow \infty \), and as \( x \rightarrow-\infty, f(x) \rightarrow \infty \)"
This corresponds to the second choice:
"As [tex]\( x \rightarrow \infty, f(x) \rightarrow \infty \)[/tex], and as [tex]\( x \rightarrow-\infty, f(x) \rightarrow \infty \)[/tex]"
[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]
From the table, we observe the following:
1. As \( x \) increases from \(-5\) to \(3\), the function \( f(x) \) starts at \( 14 \), decreases to \( -6 \) around \( x = -1 \), and then increases back to \( 6 \) as \( x \) reaches \( 3 \).
2. This pattern suggests that \( f(x) \) tends to increase again as \( x \) continues to grow positively.
3. Similarly, If we consider \( x \) becoming more negative from \(-5\) onwards, \( f(x) \) is seen to start high at \( 14 \), decrease significantly, and shows a potential to start increasing beyond the given negative values.
Based on these observations, the trend indicates:
- As \( x \rightarrow \infty \), \( f(x) \rightarrow \infty \)
- As \( x \rightarrow -\infty \), \( f(x) \rightarrow \infty \)
So, the statement that best describes the prediction for the end behavior of the graph of \( f(x) \) is:
"As \( x \rightarrow \infty, f(x) \rightarrow \infty \), and as \( x \rightarrow-\infty, f(x) \rightarrow \infty \)"
This corresponds to the second choice:
"As [tex]\( x \rightarrow \infty, f(x) \rightarrow \infty \)[/tex], and as [tex]\( x \rightarrow-\infty, f(x) \rightarrow \infty \)[/tex]"
