Answer :
To determine whether a relation represented by a table is a function, we need to ensure that each input has a unique output. For a relation to be a function, each unique input value should map to exactly one output value.
Let's analyze each given table one by one.
1. First Table:
[tex]\[ \begin{tabular}{c|cccc} Input & -3 & 5 & 7 & 4 \\ \hline Output & 4 & -3 & 1 & 4 \end{tabular} \][/tex]
- The input values are [tex]\(-3, 5, 7, 4\)[/tex].
- The output values are [tex]\(4, -3, 1, 4\)[/tex].
Each input maps to a unique output ([tex]\(-3 \rightarrow 4\)[/tex], [tex]\(5 \rightarrow -3\)[/tex], [tex]\(7 \rightarrow 1\)[/tex], [tex]\(4 \rightarrow 4\)[/tex]). There are no repeated input values with different outputs. Therefore, this table represents a function.
2. Second Table:
[tex]\[ \begin{tabular}{c|cccc} Input & -3 & 1 & 9 & 3 \\ \hline Output & 8 & -1 & -4 & -4 \end{tabular} \][/tex]
- The input values are [tex]\(-3, 1, 9, 3\)[/tex].
- The output values are [tex]\(8, -1, -4, -4\)[/tex].
Each input maps to a unique output ([tex]\(-3 \rightarrow 8\)[/tex], [tex]\(1 \rightarrow -1\)[/tex], [tex]\(9 \rightarrow -4\)[/tex], [tex]\(3 \rightarrow -4\)[/tex]). There are no repeated input values with different outputs. Therefore, this table represents a function.
3. Third Table:
[tex]\[ \begin{tabular}{c|cccc} Input & -3 & 7 & -5 & -2 \\ \hline Output & 13 & 10 & 11 & 11 \end{tabular} \][/tex]
- The input values are [tex]\(-3, 7, -5, -2\)[/tex].
- The output values are [tex]\(13, 10, 11, 11\)[/tex].
Each input maps to a unique output ([tex]\(-3 \rightarrow 13\)[/tex], [tex]\(7 \rightarrow 10\)[/tex], [tex]\(-5 \rightarrow 11\)[/tex], [tex]\(-2 \rightarrow 11\)[/tex]). There are no repeated input values with different outputs. Therefore, this table represents a function.
4. Fourth Table:
[tex]\[ \begin{tabular}{c|cccc} Input & 0 & 3 & 0 & 6 \\ \hline Output & 0 & 0 & 0 & 0 \end{tabular} \][/tex]
- The input values are [tex]\(0, 3, 0, 6\)[/tex].
- The output values are [tex]\(0, 0, 0, 0\)[/tex].
Each input value [tex]\(3\rightarrow 0\)[/tex], [tex]\(6\rightarrow 0\)[/tex] and [tex]\(0 \rightarrow 0\)[/tex] each time. There are no repeated input values with different outputs. Therefore, this table represents a function.
On evaluating all the tables, we can conclude that the relations in all the given tables are functions. Therefore, the correct selections are all four tables: 1, 2, 3, and 4.
Let's analyze each given table one by one.
1. First Table:
[tex]\[ \begin{tabular}{c|cccc} Input & -3 & 5 & 7 & 4 \\ \hline Output & 4 & -3 & 1 & 4 \end{tabular} \][/tex]
- The input values are [tex]\(-3, 5, 7, 4\)[/tex].
- The output values are [tex]\(4, -3, 1, 4\)[/tex].
Each input maps to a unique output ([tex]\(-3 \rightarrow 4\)[/tex], [tex]\(5 \rightarrow -3\)[/tex], [tex]\(7 \rightarrow 1\)[/tex], [tex]\(4 \rightarrow 4\)[/tex]). There are no repeated input values with different outputs. Therefore, this table represents a function.
2. Second Table:
[tex]\[ \begin{tabular}{c|cccc} Input & -3 & 1 & 9 & 3 \\ \hline Output & 8 & -1 & -4 & -4 \end{tabular} \][/tex]
- The input values are [tex]\(-3, 1, 9, 3\)[/tex].
- The output values are [tex]\(8, -1, -4, -4\)[/tex].
Each input maps to a unique output ([tex]\(-3 \rightarrow 8\)[/tex], [tex]\(1 \rightarrow -1\)[/tex], [tex]\(9 \rightarrow -4\)[/tex], [tex]\(3 \rightarrow -4\)[/tex]). There are no repeated input values with different outputs. Therefore, this table represents a function.
3. Third Table:
[tex]\[ \begin{tabular}{c|cccc} Input & -3 & 7 & -5 & -2 \\ \hline Output & 13 & 10 & 11 & 11 \end{tabular} \][/tex]
- The input values are [tex]\(-3, 7, -5, -2\)[/tex].
- The output values are [tex]\(13, 10, 11, 11\)[/tex].
Each input maps to a unique output ([tex]\(-3 \rightarrow 13\)[/tex], [tex]\(7 \rightarrow 10\)[/tex], [tex]\(-5 \rightarrow 11\)[/tex], [tex]\(-2 \rightarrow 11\)[/tex]). There are no repeated input values with different outputs. Therefore, this table represents a function.
4. Fourth Table:
[tex]\[ \begin{tabular}{c|cccc} Input & 0 & 3 & 0 & 6 \\ \hline Output & 0 & 0 & 0 & 0 \end{tabular} \][/tex]
- The input values are [tex]\(0, 3, 0, 6\)[/tex].
- The output values are [tex]\(0, 0, 0, 0\)[/tex].
Each input value [tex]\(3\rightarrow 0\)[/tex], [tex]\(6\rightarrow 0\)[/tex] and [tex]\(0 \rightarrow 0\)[/tex] each time. There are no repeated input values with different outputs. Therefore, this table represents a function.
On evaluating all the tables, we can conclude that the relations in all the given tables are functions. Therefore, the correct selections are all four tables: 1, 2, 3, and 4.
