To determine which data sets represent a function, we need to understand the definition of a function: a relation in which each input (or x-value) is associated with exactly one output (or y-value).
Here's a step-by-step check for each data set:
1. First data set:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-5 & 10 \\
\hline
-3 & 5 \\
\hline
-3 & 4 \\
\hline
0 & 0 \\
\hline
5 & -10 \\
\hline
\end{array}
\][/tex]
- We have x-values: -5, -3, -3, 0, and 5.
- The x-value -3 is repeated and associated with different y-values (5 and 4).
- Hence, this set does not represent a function.
2. Second data set:
[tex]\[
\{(-8, -2), (-4, 1), (0, -2), (2, 3), (4, -4)\}
\][/tex]
- We have x-values: -8, -4, 0, 2, and 4.
- Each x-value appears only once with a unique y-value associated.
- Hence, this set represents a function.
3. Third data set:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & -3 \\
\hline
-1 & -2 \\
\hline
0 & -1 \\
\hline
0 & 0 \\
\hline
1 & -1 \\
\hline
\end{array}
\][/tex]
- We have x-values: -2, -1, 0, 0, and 1.
- The x-value 0 is repeated and is associated with different y-values (-1 and 0).
- Hence, this set does not represent a function.
4. Fourth data set:
[tex]\[
\{(-12, 4), (-6, 10), (-4, 15), (-8, 18), (-12, 24)\}
\][/tex]
- We have x-values: -12, -6, -4, -8, and -12.
- The x-value -12 is repeated and associated with different y-values (4 and 24).
- Hence, this set does not represent a function.
In summary, out of the given sets, only the second data set represents a function.
