To determine which of the given sets of ordered pairs represents a function, we need to ensure that each input (or [tex]\(x\)[/tex]-value) has exactly one output (or [tex]\(y\)[/tex]-value). A relation is a function if and only if no [tex]\(x\)[/tex]-value is repeated with different [tex]\(y\)[/tex]-values.
Let's analyze each set of ordered pairs:
1. Set 1: [tex]\(\{(7,3),(-6,8),(-3,5),(0,-3),(7,11)\}\)[/tex]
- We have the pairs: (7,3), (-6,8), (-3,5), (0,-3), (7,11).
- The [tex]\(x\)[/tex]-value 7 appears twice but with different [tex]\(y\)[/tex]-values (3 and 11).
- Since 7 maps to two different [tex]\(y\)[/tex]-values, this set does not represent a function.
2. Set 2: [tex]\(\{(5,2),(-4,2),(3,6),(0,4),(-1,2)\}\)[/tex]
- We have the pairs: (5,2), (-4,2), (3,6), (0,4), (-1,2).
- Each [tex]\(x\)[/tex]-value (5, -4, 3, 0, -1) is unique and maps to one specific [tex]\(y\)[/tex]-value.
- Since no [tex]\(x\)[/tex]-value repeats with different [tex]\(y\)[/tex]-values, this set represents a function.
3. Set 3: [tex]\(\{(5,4),(5,6),(5,8),(5,10),(5,12)\}\)[/tex]
- We have the pairs: (5,4), (5,6), (5,8), (5,10), (5,12).
- The [tex]\(x\)[/tex]-value 5 appears multiple times with different [tex]\(y\)[/tex]-values (4, 6, 8, 10, 12).
- Since 5 maps to multiple [tex]\(y\)[/tex]-values, this set does not represent a function.
4. Set 4: [tex]\(\{(-3,-2),(-2,-1),(0,-1),(0,1),(1,2)\}\)[/tex]
- We have the pairs: (-3,-2), (-2,-1), (0,-1), (0,1), (1,2).
- The [tex]\(x\)[/tex]-value 0 appears twice with different [tex]\(y\)[/tex]-values (-1 and 1).
- Since 0 maps to two different [tex]\(y\)[/tex]-values, this set does not represent a function.
Based on this analysis, only the second set of ordered pairs represents a function. Therefore, the relation given by:
[tex]\[
\{(5,2),(-4,2),(3,6),(0,4),(-1,2)\}
\][/tex]
is a function.
