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Sagot :
To solve this problem, we must determine the domain and range of the given relation and then decide if it is a function.
Let's break down the ordered pairs in the relation:
[tex]\[ R = \{(-17, -4), (7, -8), (20, 16), (-1, -19), (15, -3)\} \][/tex]
### Domain
The domain of a relation is the set of all first elements (or x-values) from the ordered pairs. From the given relation, we extract the first elements:
- From [tex]\((-17, -4)\)[/tex], the first element is [tex]\(-17\)[/tex].
- From [tex]\((7, -8)\)[/tex], the first element is [tex]\(7\)[/tex].
- From [tex]\((20, 16)\)[/tex], the first element is [tex]\(20\)[/tex].
- From [tex]\((-1, -19)\)[/tex], the first element is [tex]\(-1\)[/tex].
- From [tex]\((15, -3)\)[/tex], the first element is [tex]\(15\)[/tex].
Thus, the domain is:
[tex]\[ \{-17, 7, 20, -1, 15\} \][/tex]
### Range
The range of a relation is the set of all second elements (or y-values) from the ordered pairs. From the given relation, we extract the second elements:
- From [tex]\((-17, -4)\)[/tex], the second element is [tex]\(-4\)[/tex].
- From [tex]\((7, -8)\)[/tex], the second element is [tex]\(-8\)[/tex].
- From [tex]\((20, 16)\)[/tex], the second element is [tex]\(16\)[/tex].
- From [tex]\((-1, -19)\)[/tex], the second element is [tex]\(-19\)[/tex].
- From [tex]\((15, -3)\)[/tex], the second element is [tex]\(-3\)[/tex].
Thus, the range is:
[tex]\[ \{-19, 16, -8, -4, -3\} \][/tex]
### Function Check
A relation is a function if every element in the domain maps to exactly one element in the range. In simpler terms, each x-value should have exactly one corresponding y-value.
Given the ordered pairs in the relation [tex]\( R \)[/tex], we see that:
- [tex]\(-17\)[/tex] maps to [tex]\(-4\)[/tex].
- [tex]\(7\)[/tex] maps to [tex]\(-8\)[/tex].
- [tex]\(20\)[/tex] maps to [tex]\(16\)[/tex].
- [tex]\(-1\)[/tex] maps to [tex]\(-19\)[/tex].
- [tex]\(15\)[/tex] maps to [tex]\(-3\)[/tex].
Each x-value has a unique y-value, meaning no x-value is repeated with a different y-value. Therefore, this relation is indeed a function.
### Summary
Based on the analysis, the domain, range, and function determination can be stated as follows:
Domain: [tex]\(\{-17, 7, 20, -1, 15\}\)[/tex]
Range: [tex]\(\{-19, 16, -8, -4, -3\}\)[/tex]
Is the relation [tex]\( R \)[/tex] a function?
Yes
Let's break down the ordered pairs in the relation:
[tex]\[ R = \{(-17, -4), (7, -8), (20, 16), (-1, -19), (15, -3)\} \][/tex]
### Domain
The domain of a relation is the set of all first elements (or x-values) from the ordered pairs. From the given relation, we extract the first elements:
- From [tex]\((-17, -4)\)[/tex], the first element is [tex]\(-17\)[/tex].
- From [tex]\((7, -8)\)[/tex], the first element is [tex]\(7\)[/tex].
- From [tex]\((20, 16)\)[/tex], the first element is [tex]\(20\)[/tex].
- From [tex]\((-1, -19)\)[/tex], the first element is [tex]\(-1\)[/tex].
- From [tex]\((15, -3)\)[/tex], the first element is [tex]\(15\)[/tex].
Thus, the domain is:
[tex]\[ \{-17, 7, 20, -1, 15\} \][/tex]
### Range
The range of a relation is the set of all second elements (or y-values) from the ordered pairs. From the given relation, we extract the second elements:
- From [tex]\((-17, -4)\)[/tex], the second element is [tex]\(-4\)[/tex].
- From [tex]\((7, -8)\)[/tex], the second element is [tex]\(-8\)[/tex].
- From [tex]\((20, 16)\)[/tex], the second element is [tex]\(16\)[/tex].
- From [tex]\((-1, -19)\)[/tex], the second element is [tex]\(-19\)[/tex].
- From [tex]\((15, -3)\)[/tex], the second element is [tex]\(-3\)[/tex].
Thus, the range is:
[tex]\[ \{-19, 16, -8, -4, -3\} \][/tex]
### Function Check
A relation is a function if every element in the domain maps to exactly one element in the range. In simpler terms, each x-value should have exactly one corresponding y-value.
Given the ordered pairs in the relation [tex]\( R \)[/tex], we see that:
- [tex]\(-17\)[/tex] maps to [tex]\(-4\)[/tex].
- [tex]\(7\)[/tex] maps to [tex]\(-8\)[/tex].
- [tex]\(20\)[/tex] maps to [tex]\(16\)[/tex].
- [tex]\(-1\)[/tex] maps to [tex]\(-19\)[/tex].
- [tex]\(15\)[/tex] maps to [tex]\(-3\)[/tex].
Each x-value has a unique y-value, meaning no x-value is repeated with a different y-value. Therefore, this relation is indeed a function.
### Summary
Based on the analysis, the domain, range, and function determination can be stated as follows:
Domain: [tex]\(\{-17, 7, 20, -1, 15\}\)[/tex]
Range: [tex]\(\{-19, 16, -8, -4, -3\}\)[/tex]
Is the relation [tex]\( R \)[/tex] a function?
Yes
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