To determine which of the given sets represent valid functions, we need to check if each set adheres to the definition of a function. Specifically, a function from a set [tex]\(A\)[/tex] to a set [tex]\(B\)[/tex] assigns exactly one element of [tex]\(B\)[/tex] to each element of [tex]\(A\)[/tex]. This means that in the context of pairs [tex]\((x, y)\)[/tex], every [tex]\(x\)[/tex]-value should be associated with only one [tex]\(y\)[/tex]-value.
Let's analyze each given set:
1. Set [tex]\(S = \{(-3, -5), (1, 2), (4, 6), (-3, -5), (12, 16)\}:
- Extract the \(x\)[/tex]-values: [tex]\(-3, 1, 4, \(-3\)[/tex], 12.
- The [tex]\(x = -3\)[/tex] appears twice with the same [tex]\(y\)[/tex]-value [tex]\(-5\)[/tex], so each [tex]\(x\)[/tex] associates with exactly one [tex]\(y\)[/tex].
- Thus, [tex]\(S\)[/tex] represents a valid function.
2. Set [tex]\(G = \{(0, -1), (1, 3), (1, 5), (8, 9), (12, 16)\}:
- Extract the \(x\)[/tex]-values: [tex]\(0, 1, 1, 8, 12\)[/tex].
- The [tex]\(x = 1\)[/tex] is associated with two different [tex]\(y\)[/tex]-values [tex]\(3\)[/tex] and [tex]\(5\)[/tex], which violates the definition of a function.
- Thus, [tex]\(G\)[/tex] does not represent a valid function.
3. Set [tex]\(R = \{(-3, -4), (1, 2), (5, 4), (7, 9), (13, 16)\}:
- Extract the \(x\)[/tex]-values: [tex]\(-3, 1, 5, 7, 13\)[/tex].
- Each [tex]\(x\)[/tex]-value is unique.
- Thus, [tex]\(R\)[/tex] represents a valid function.
4. Set [tex]\(F = \{(-1, 0), (1, 2), (6, 5), (8, 9), (12, 14)\}:
- Extract the \(x\)[/tex]-values: [tex]\(-1, 1, 6, 8, 12\)[/tex].
- Each [tex]\(x\)[/tex]-value is unique.
- Thus, [tex]\(F\)[/tex] represents a valid function.
After analyzing all the sets, we can conclude that the data sets [tex]\(S\)[/tex], [tex]\(R\)[/tex], and [tex]\(F\)[/tex] represent valid functions. Therefore, the sets [tex]\(S\)[/tex], [tex]\(R\)[/tex], and [tex]\(F\)[/tex] are the valid functions, and their corresponding indexes are:
[tex]\[ \boxed{1, 3, 4} \][/tex]
