To determine which of the given relations is a function, we need to check whether each set of ordered pairs satisfies the definition of a function. Specifically, a relation is a function if every input [tex]\( x \)[/tex] relates to exactly one output [tex]\( y \)[/tex]. This means that no [tex]\( x \)[/tex]-value should be repeated with a different [tex]\( y \)[/tex]-value.
Let’s examine each set:
1. [tex]\(\{(-2,9),(3,8),(7,2),(-2,13)\}\)[/tex]:
- The [tex]\( x \)[/tex]-values are [tex]\(-2, 3, 7, -2\)[/tex].
- [tex]\( x = -2 \)[/tex] is repeated.
- Therefore, this relation is not a function.
2. [tex]\(\{(6,1),(1,2),(-1,2),(0,-3)\}\)[/tex]:
- The [tex]\( x \)[/tex]-values are [tex]\(6, 1, -1, 0\)[/tex].
- No [tex]\( x \)[/tex]-value is repeated.
- Therefore, this relation is a function.
3. [tex]\(\{(-2,-4),(7,6),(7,12),(1,4)\}\)[/tex]:
- The [tex]\( x \)[/tex]-values are [tex]\(-2, 7, 7, 1\)[/tex].
- [tex]\( x = 7 \)[/tex] is repeated.
- Therefore, this relation is not a function.
4. [tex]\(\{(5,-3),(2,13),(1,-1),(1,-5)\}\)[/tex]:
- The [tex]\( x \)[/tex]-values are [tex]\(5, 2, 1, 1\)[/tex].
- [tex]\( x = 1 \)[/tex] is repeated.
- Therefore, this relation is not a function.
5. [tex]\(\{(9,1),(7,13),(3,1),(5,11)\}\)[/tex]:
- The [tex]\( x \)[/tex]-values are [tex]\(9, 7, 3, 5\)[/tex].
- No [tex]\( x \)[/tex]-value is repeated.
- Therefore, this relation is a function.
The relations that are functions are:
[tex]\[
\{(6,1),(1,2),(-1,2),(0,-3)\}
\][/tex]
[tex]\[
\{(9,1),(7,13),(3,1),(5,11)\}
\][/tex]
Thus, the correct answer choices are:
[tex]\[
\boxed{\{(6,1),(1,2),(-1,2),(0,-3)\}}
\][/tex]
[tex]\[
\boxed{\{(9,1),(7,13),(3,1),(5,11)\}}
\][/tex]
