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Sagot :
To determine which of the given sets of ordered pairs does not represent a function, we need to recall the definition of a function. In mathematics, a function is defined such that each input (often denoted as [tex]\(x\)[/tex]) is mapped to exactly one output (often denoted as [tex]\(y\)[/tex]). This means that no two different ordered pairs can have the same first component (the [tex]\(x\)[/tex]-value) but different second components (the [tex]\(y\)[/tex]-values).
Let's analyze each set of ordered pairs to check whether they meet this criterion.
1. Set 1: [tex]\(\{(2,-7), (-1,-4), (-7,-6), (-7,9)\}\)[/tex]
- Here we have pairs [tex]\((2, -7)\)[/tex], [tex]\((-1, -4)\)[/tex], [tex]\((-7, -6)\)[/tex], and [tex]\((-7, 9)\)[/tex].
- Notice that the [tex]\(x\)[/tex]-value [tex]\(-7\)[/tex] appears in two different pairs: [tex]\((-7, -6)\)[/tex] and [tex]\((-7, 9)\)[/tex].
- Since [tex]\(-7\)[/tex] maps to both [tex]\(-6\)[/tex] and [tex]\(9\)[/tex], this violates the definition of a function.
- Therefore, this set does not represent a function.
2. Set 2: [tex]\(\{(3,3), (8,-9), (-6,3), (-8,4)\}\)[/tex]
- Here we have pairs [tex]\((3, 3)\)[/tex], [tex]\((8, -9)\)[/tex], [tex]\((-6, 3)\)[/tex], and [tex]\((-8, 4)\)[/tex].
- All [tex]\(x\)[/tex]-values are unique: [tex]\(3\)[/tex], [tex]\(8\)[/tex], [tex]\(-6\)[/tex], [tex]\(-8\)[/tex].
- Each [tex]\(x\)[/tex]-value maps to exactly one [tex]\(y\)[/tex]-value.
- Thus, this set represents a function.
3. Set 3: [tex]\(\{(-7,9), (3,9), (7,7), (1,2)\}\)[/tex]
- Here we have pairs [tex]\((-7, 9)\)[/tex], [tex]\((3, 9)\)[/tex], [tex]\((7, 7)\)[/tex], and [tex]\((1, 2)\)[/tex].
- All [tex]\(x\)[/tex]-values are unique: [tex]\(-7\)[/tex], [tex]\(3\)[/tex], [tex]\(7\)[/tex], [tex]\(1\)[/tex].
- Each [tex]\(x\)[/tex]-value maps to exactly one [tex]\(y\)[/tex]-value.
- Thus, this set represents a function.
4. Set 4: [tex]\(\{(1,0), (-7,-3), (8,6), (-9,6)\}\)[/tex]
- Here we have pairs [tex]\((1, 0)\)[/tex], [tex]\((-7, -3)\)[/tex], [tex]\((8, 6)\)[/tex], and [tex]\((-9, 6)\)[/tex].
- All [tex]\(x\)[/tex]-values are unique: [tex]\(1\)[/tex], [tex]\(-7\)[/tex], [tex]\(8\)[/tex], [tex]\(-9\)[/tex].
- Each [tex]\(x\)[/tex]-value maps to exactly one [tex]\(y\)[/tex]-value.
- Thus, this set represents a function.
Based on these analyses, the set that does not represent a function is:
[tex]\[ \boxed{\{(2,-7),(-1,-4),(-7,-6),(-7,9)\}} \][/tex]
Let's analyze each set of ordered pairs to check whether they meet this criterion.
1. Set 1: [tex]\(\{(2,-7), (-1,-4), (-7,-6), (-7,9)\}\)[/tex]
- Here we have pairs [tex]\((2, -7)\)[/tex], [tex]\((-1, -4)\)[/tex], [tex]\((-7, -6)\)[/tex], and [tex]\((-7, 9)\)[/tex].
- Notice that the [tex]\(x\)[/tex]-value [tex]\(-7\)[/tex] appears in two different pairs: [tex]\((-7, -6)\)[/tex] and [tex]\((-7, 9)\)[/tex].
- Since [tex]\(-7\)[/tex] maps to both [tex]\(-6\)[/tex] and [tex]\(9\)[/tex], this violates the definition of a function.
- Therefore, this set does not represent a function.
2. Set 2: [tex]\(\{(3,3), (8,-9), (-6,3), (-8,4)\}\)[/tex]
- Here we have pairs [tex]\((3, 3)\)[/tex], [tex]\((8, -9)\)[/tex], [tex]\((-6, 3)\)[/tex], and [tex]\((-8, 4)\)[/tex].
- All [tex]\(x\)[/tex]-values are unique: [tex]\(3\)[/tex], [tex]\(8\)[/tex], [tex]\(-6\)[/tex], [tex]\(-8\)[/tex].
- Each [tex]\(x\)[/tex]-value maps to exactly one [tex]\(y\)[/tex]-value.
- Thus, this set represents a function.
3. Set 3: [tex]\(\{(-7,9), (3,9), (7,7), (1,2)\}\)[/tex]
- Here we have pairs [tex]\((-7, 9)\)[/tex], [tex]\((3, 9)\)[/tex], [tex]\((7, 7)\)[/tex], and [tex]\((1, 2)\)[/tex].
- All [tex]\(x\)[/tex]-values are unique: [tex]\(-7\)[/tex], [tex]\(3\)[/tex], [tex]\(7\)[/tex], [tex]\(1\)[/tex].
- Each [tex]\(x\)[/tex]-value maps to exactly one [tex]\(y\)[/tex]-value.
- Thus, this set represents a function.
4. Set 4: [tex]\(\{(1,0), (-7,-3), (8,6), (-9,6)\}\)[/tex]
- Here we have pairs [tex]\((1, 0)\)[/tex], [tex]\((-7, -3)\)[/tex], [tex]\((8, 6)\)[/tex], and [tex]\((-9, 6)\)[/tex].
- All [tex]\(x\)[/tex]-values are unique: [tex]\(1\)[/tex], [tex]\(-7\)[/tex], [tex]\(8\)[/tex], [tex]\(-9\)[/tex].
- Each [tex]\(x\)[/tex]-value maps to exactly one [tex]\(y\)[/tex]-value.
- Thus, this set represents a function.
Based on these analyses, the set that does not represent a function is:
[tex]\[ \boxed{\{(2,-7),(-1,-4),(-7,-6),(-7,9)\}} \][/tex]
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