To determine which of the given sets of ordered pairs represents a function, we need to check if each set fulfills the criteria for being a function. A function assigns exactly one output [tex]\( y \)[/tex] for each input [tex]\( x \)[/tex]. This means that for each unique [tex]\( x \)[/tex] value, there should be a unique corresponding [tex]\( y \)[/tex] value.
Set 1: [tex]\(\{(-6,-5),(-4,-3),(-2,0),(-2,2),(0,4)\}\)[/tex]
- The [tex]\( x \)[/tex]-values are: [tex]\(-6, -4, -2, -2, 0\)[/tex].
- The [tex]\( x \)[/tex]-value [tex]\(-2\)[/tex] appears twice with different [tex]\( y \)[/tex]-values: [tex]\((\,-2,0\,)\)[/tex] and [tex]\((\,-2,2\,)\)[/tex].
Since [tex]\(-2\)[/tex] maps to both [tex]\( 0 \)[/tex] and [tex]\( 2 \)[/tex], this set does not represent a function.
Set 2: [tex]\(\{(-5,-5),(-5,-4),(-5,-3),(-5,-2),(-5,0)\}\)[/tex]
- The [tex]\( x \)[/tex]-values are: [tex]\(-5, -5, -5, -5, -5\)[/tex].
- The [tex]\( x \)[/tex]-value [tex]\(-5\)[/tex] appears multiple times with different [tex]\( y \)[/tex]-values: [tex]\((\,-5,-5\,)\)[/tex], [tex]\((\,-5,-4\,)\)[/tex], [tex]\((\,-5,-3\,)\)[/tex], [tex]\((\,-5,-2\,)\)[/tex], and [tex]\((\,-5,0\,)\)[/tex].
Since [tex]\(-5\)[/tex] maps to multiple [tex]\( y \)[/tex]-values ([tex]\(-5, -4, -3, -2, 0\)[/tex]), this set does not represent a function.
Set 3: [tex]\(\{(-4,-5),(-3,0),(-2,-4),(0,-3),(2,-2)\}\)[/tex]
- The [tex]\( x \)[/tex]-values are: [tex]\(-4, -3, -2, 0, 2\)[/tex].
- Each [tex]\( x \)[/tex]-value appears exactly once and has a unique corresponding [tex]\( y \)[/tex]-value.
Since each [tex]\( x \)[/tex]-value corresponds to exactly one [tex]\( y \)[/tex]-value, this set represents a function.
Set 4: [tex]\(\{(-6,-3),(-6,-2),(-5,-3),(-3,-3),(0,0)\}\)[/tex]
- The [tex]\( x \)[/tex]-values are: [tex]\(-6, -6, -5, -3, 0\)[/tex].
- The [tex]\( x \)[/tex]-value [tex]\(-6\)[/tex] appears twice with different [tex]\( y \)[/tex]-values: [tex]\((\,-6,-3\,)\)[/tex] and [tex]\((\,-6,-2\,)\)[/tex].
Since [tex]\(-6\)[/tex] maps to both [tex]\(-3\)[/tex] and [tex]\(-2\)[/tex], this set does not represent a function.
Conclusion:
Among the given sets, only Set 3 satisfies the condition required for a function, where each [tex]\( x \)[/tex]-value has a unique corresponding [tex]\( y \)[/tex]-value.
Thus, the set [tex]\(\{(-4,-5), (-3,0), (-2,-4), (0,-3), (2,-2)\}\)[/tex] represents a function.
