To determine which table represents a function, we need to ensure that for every unique input value [tex]\( x \)[/tex], there is only one corresponding output value [tex]\( y \)[/tex]. This means that each input value should map to exactly one output value.
Let's go through the tables one by one to check this property:
### Table 1
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Input (x)} & -4 & -2 & -2 & -1 & 0 \\
\hline
\text{Output (y)} & 6 & 8 & 10 & 12 & 14 \\
\hline
\end{array}
\][/tex]
- The input value [tex]\(-2\)[/tex] appears twice, with different outputs ([tex]\(8\)[/tex] and [tex]\(10\)[/tex]).
- This violates the definition of a function.
Table 1 does NOT represent a function.
### Table 2
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Input (x)} & -2 & 0 & 2 & 4 & 6 \\
\hline
\text{Output (y)} & 3 & 3 & 6 & 6 & 9 \\
\hline
\end{array}
\][/tex]
- Each input value [tex]\(-2, 0, 2, 4, 6\)[/tex] is unique and maps to one specific output value.
- There are no repeated input values.
Table 2 represents a function.
### Table 3
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Input (x)} & 10 & 20 & 30 & 30 & 40 \\
\hline
\text{Output (y)} & -5 & -6 & -7 & -8 & -9 \\
\hline
\end{array}
\][/tex]
- The input value [tex]\(30\)[/tex] appears twice, with different outputs ([tex]\(-7\)[/tex] and [tex]\(-8\)[/tex]).
- This violates the definition of a function.
Table 3 does NOT represent a function.
### Table 4
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Input (x)} & -5 & -5 & -5 & -5 & -5 \\
\hline
\text{Output (y)} & -5 & 0 & 5 & 10 & 15 \\
\hline
\end{array}
\][/tex]
- The input value [tex]\(-5\)[/tex] appears multiple times with different outputs ([tex]\(-5, 0, 5, 10, 15\)[/tex]).
- This violates the definition of a function.
Table 4 does NOT represent a function.
Conclusion:
Among the given tables, only Table 2 represents a function.
