To determine whether the given relation is a function, we need to understand the definition of a function.
A function is a relation in which every element of the domain (the set of all possible inputs, or 'x' values) is associated with exactly one element of the codomain (the set of all possible outputs, or 'y' values). This means that for every 'x' value in the relation, there should be only one corresponding 'y' value.
Given the relation:
[tex]\[
\{(3,-2),(1,2),(-1,-4),(-1,2)\}
\][/tex]
We can list the pairs of 'x' and 'y' values:
[tex]\[
(3, -2), (1, 2), (-1, -4), (-1, 2)
\][/tex]
Next, observe the 'x' values in these pairs:
[tex]\[
3, 1, -1, -1
\][/tex]
We see that the 'x' value [tex]\(-1\)[/tex] appears more than once, and it corresponds to different 'y' values, specifically [tex]\(-4\)[/tex] and [tex]\(2\)[/tex].
This situation violates the definition of a function because a single 'x' value must correspond to only one 'y' value. Since the 'x' value [tex]\(-1\)[/tex] corresponds to multiple 'y' values, the given relation is not a function.
Thus, the answer to the question is:
[tex]\[
\text{No}
\][/tex]
