To determine the domain of a function, we need to identify all possible input values, or [tex]\(x\)[/tex]-values, that the function can take. The domain is the set of these [tex]\(x\)[/tex]-values.
Let's examine the given table of values:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-2 & 0 \\
\hline
-1 & 1 \\
\hline
0 & 2 \\
\hline
1 & 3 \\
\hline
\end{tabular}
\][/tex]
From the table, we can see that the function has the following [tex]\(x\)[/tex]-values:
- [tex]\(x = -2\)[/tex]
- [tex]\(x = -1\)[/tex]
- [tex]\(x = 0\)[/tex]
- [tex]\(x = 1\)[/tex]
These values make up the domain of the function. Now, let us compare these [tex]\(x\)[/tex]-values with each of the given options:
A. [tex]\(\{0, 1, 2, 3\}\)[/tex]
- This set contains different values, which are not part of our identified domain.
B. [tex]\((-2, 0), (-1, 1), (0, 2), (1, 3)\)[/tex]
- This option pairs each [tex]\(x\)[/tex]-value with its corresponding [tex]\(y\)[/tex]-value, but the domain should only include the [tex]\(x\)[/tex]-values.
C. [tex]\(\{-2, -1, 0, 1\}\)[/tex]
- This set matches exactly with our identified domain.
D. [tex]\(\{-2, -1, 0, 1, 2, 3\}\)[/tex]
- This set includes extra values (2 and 3) that are not part of our identified domain.
Thus, the correct answer is:
C. [tex]\(\{-2, -1, 0, 1\}\)[/tex]
The domain of the function shown in the table is [tex]\(\{-2, -1, 0, 1\}\)[/tex].
