To determine which relation is a function of [tex]\( x \)[/tex], we need to check that for each value of [tex]\( x \)[/tex], there is only one unique value of [tex]\( y \)[/tex]. In other words, no [tex]\( x \)[/tex]-value is repeated with a different [tex]\( y \)[/tex]-value.
Let's analyze each set of relations one by one:
### 1. First Relation
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-1 & 7 \\
\hline
2 & -9 \\
\hline
2 & 8 \\
\hline
3 & -4 \\
\hline
\end{array}
\][/tex]
- Here, the [tex]\( x \)[/tex]-value 2 corresponds to both [tex]\( y = -9 \)[/tex] and [tex]\( y = 8 \)[/tex].
- Because the [tex]\( x \)[/tex]-value 2 maps to two different [tex]\( y \)[/tex]-values, this relation is not a function.
### 2. Second Relation
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-8 & -9 \\
\hline
-8 & 2 \\
\hline
1 & -9 \\
\hline
1 & 2 \\
\hline
\end{array}
\][/tex]
- Here, the [tex]\( x \)[/tex]-value -8 corresponds to both [tex]\( y = -9 \)[/tex] and [tex]\( y = 2 \)[/tex].
- Additionally, the [tex]\( x \)[/tex]-value 1 corresponds to both [tex]\( y = -9 \)[/tex] and [tex]\( y = 2 \)[/tex].
- Because there are repeated [tex]\( x \)[/tex]-values with different [tex]\( y \)[/tex]-values, this relation is not a function.
### 3. Third Relation
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-5 & 1 \\
\hline
-5 & 7 \\
\hline
-5 & -9 \\
\hline
-5 & 2 \\
\hline
\end{array}
\][/tex]
- Here, the [tex]\( x \)[/tex]-value -5 corresponds to four different [tex]\( y \)[/tex]-values: 1, 7, -9, and 2.
- Because the [tex]\( x \)[/tex]-value -5 maps to multiple [tex]\( y \)[/tex]-values, this relation is not a function.
### 4. Fourth Relation
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 1 \\
\hline
\end{array}
\][/tex]
- Here, the [tex]\( x \)[/tex]-value 2 corresponds to one [tex]\( y \)[/tex]-value: 1.
- Since there is only one [tex]\( x \)[/tex]-value and it maps to only one [tex]\( y \)[/tex]-value, this relation is a function.
### Conclusion
The relation that is a function of [tex]\( x \)[/tex] is the fourth relation:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 1 \\
\hline
\end{array}
\][/tex]
