To determine the [tex]\( x \)[/tex]-intercepts given the table of values for the function [tex]\( f(x) \)[/tex], we need to identify the points where the function crosses the [tex]\( x \)[/tex]-axis, that is, where [tex]\( f(x) = 0 \)[/tex].
Here is the table provided:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
2 & 20 \\
\hline
-1 & 0 \\
\hline
0 & -6 \\
\hline
1 & -4 \\
\hline
2 & 0 \\
\hline
3 & 0 \\
\hline
\end{array}
\][/tex]
Step-by-Step Solution:
1. Identify [tex]\( f(x) = 0 \)[/tex]:
- We need to find where [tex]\( f(x) \)[/tex] equals zero.
2. Examine each point in the table:
- For [tex]\( x = 2, \, f(x) = 20 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) \neq 0 \)[/tex].
- For [tex]\( x = -1, \, f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( (-1, 0) \)[/tex] is a candidate.
- For [tex]\( x = 0, \, f(x) = -6 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) \neq 0 \)[/tex].
- For [tex]\( x = 1, \, f(x) = -4 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) \neq 0 \)[/tex].
- For [tex]\( x = 2, \, f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( (2, 0) \)[/tex] is a candidate.
- For [tex]\( x = 3, \, f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( f(x) = 0 \)[/tex] [tex]\(\Rightarrow\)[/tex] [tex]\( (3, 0) \)[/tex] is a candidate.
Based on the points examined, the [tex]\( x \)[/tex]-intercepts, where [tex]\( f(x) = 0 \)[/tex], are:
- [tex]\( (-1, 0) \)[/tex]
- [tex]\( (2, 0) \)[/tex]
- [tex]\( (3, 0) \)[/tex]
Check given options:
- [tex]\((-1,0)\)[/tex] is one of the [tex]\( x \)[/tex]-intercepts (True).
- [tex]\((0,-6)\)[/tex] is not an [tex]\( x \)[/tex]-intercept, since [tex]\( f(x) \neq 0 \)[/tex] at this point (False).
- [tex]\((-6,0)\)[/tex] is not in the table, hence it cannot be an [tex]\( x \)[/tex]-intercept for this data (False).
- [tex]\((0,-1)\)[/tex] is not in the table, and it doesn't satisfy [tex]\( f(x) = 0 \)[/tex] either (False).
Therefore, the correct answer is:
[tex]\[ (-1, 0) \][/tex]
This is a valid [tex]\( x \)[/tex]-intercept of the continuous function based on the given data.
