Answer :
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]
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]
