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Which is an [tex]$x$[/tex]-intercept of the continuous function in the table?

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$f(x)$[/tex] \\
\hline
2 & 20 \\
\hline
-1 & 0 \\
\hline
0 & -6 \\
\hline
1 & -4 \\
\hline
2 & 0 \\
\hline
3 & 0 \\
\hline
\end{tabular}

A. [tex]$(-1, 0)$[/tex]
B. [tex]$(0, -6)$[/tex]
C. [tex]$(-6, 0)$[/tex]
D. [tex]$(0, -1)$[/tex]

Sagot :

GinnyAnswer
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.
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