To determine which of the given sets of ordered pairs represents a function, we need to recall the definition of a function. In the context of ordered pairs, a set represents a function if and only if each input (or [tex]\(x\)[/tex]-value) is associated with exactly one output (or [tex]\(y\)[/tex]-value). This means that no [tex]\(x\)[/tex]-value should be repeated with different [tex]\(y\)[/tex]-values.
We will analyze each set of ordered pairs provided in the options.
1. Set 1: [tex]\(\{(-3,-3),(-2,-2),(-1,-1),(0,0),(1,1)\}\)[/tex]
[tex]\[
\begin{aligned}
&\text{The x-values are:} \{-3, -2, -1, 0, 1\} \\
&\text{All x-values are unique.} \\
&\text{Therefore, each x-value maps to exactly one y-value.}
\end{aligned}
\][/tex]
This set represents a function.
2. Set 2: [tex]\(\{(-3,-3),(-3,-2),(-3,-1),(-3,0),(-4,-1)\}\)[/tex]
[tex]\[
\begin{aligned}
&\text{The x-values are:} \{-3, -3, -3, -3, -4\} \\
&\text{The x-value } -3 \text{ is repeated multiple times with different } y\text{-values}.
\end{aligned}
\][/tex]
This set does not represent a function because the x-value -3 is not unique; it corresponds to multiple y-values.
3. Set 3: [tex]\(\{(-3,-3),(-3,-1),(-1,-2),(-1,-1),(-1,0)\}\)[/tex]
[tex]\[
\begin{aligned}
&\text{The x-values are:} \{-3, -3, -1, -1, -1\} \\
&\text{The x-values } -3 \text{ and } -1 \text{ are repeated with different } y\text{-values}.
\end{aligned}
\][/tex]
This set does not represent a function because both -3 and -1 have multiple corresponding y-values.
4. Set 4: [tex]\(\{(-3,-3),(-3,0),(-1,-3),(0,-3),(-1,-1)\}\)[/tex]
[tex]\[
\begin{aligned}
&\text{The x-values are:} \{-3, -3, -1, 0, -1\} \\
&\text{The x-values } -3 \text{ and } -1 \text{ are repeated with different } y\text{-values}.
\end{aligned}
\][/tex]
This set does not represent a function because the x-values -3 and -1 are not unique; they appear multiple times with different y-values.
Based on the above analysis, the set of ordered pairs that represents a function is:
[tex]\[ \{(-3,-3),(-2,-2),(-1,-1),(0,0),(1,1)\} \][/tex]
So, the correct answer is the first set of ordered pairs.
