In mathematics, for a relation to be a function, each [tex]\(x\)[/tex]-value (also known as the domain) must be associated with exactly one [tex]\(y\)[/tex]-value. In other words, no two ordered pairs can have the same first element.
We are given the set of ordered pairs:
[tex]\[ \{(2, 8), (12, 3), (c, 4), (-1, 8), (0, 3)\} \][/tex]
We must determine the value of [tex]\(c\)[/tex] that ensures this relation is a function. Let's follow the steps:
1. List the [tex]\(x\)[/tex]-values: Identify the [tex]\(x\)[/tex]-values in the set of coordinates to check for any duplicates.
[tex]\[ \{2, 12, c, -1, 0\} \][/tex]
2. Check for uniqueness: For the relation to be a function, each [tex]\(x\)[/tex]-value must be unique. The current [tex]\(x\)[/tex]-values we have without [tex]\(c\)[/tex] are:
[tex]\[ \{2, 12, -1, 0\} \][/tex]
3. Determine [tex]\(c\)[/tex]: To maintain the uniqueness of the [tex]\(x\)[/tex]-values, [tex]\(c\)[/tex] must not be equal to any of the existing [tex]\(x\)[/tex]-values. This means [tex]\(c\)[/tex] should not be 2, 12, -1, or 0.
4. Choices Analysis:
- Option -1: [tex]\(c = -1\)[/tex], but -1 is already in the set.
- Option 1: [tex]\(c = 1\)[/tex], which is a unique value.
- Option 2: [tex]\(c = 2\)[/tex], but 2 is already in the set.
- Option 12: [tex]\(c = 12\)[/tex], but 12 is already in the set.
Given the options, the correct value for [tex]\(c\)[/tex] to ensure all [tex]\(x\)[/tex]-values are unique (ensuring the relation is a function) is:
[tex]\[ c = 1 \][/tex]
Thus:
[tex]\(\boxed{1}\)[/tex]
