For what value of c is the relation a function? StartSet (2, 8), (12, 3), (c, 4), (negative 1, 8), (0, 3) EndSet

Answer: To determine for which value of \( c \) the relation is a function, we need to ensure that each input, or 'x' value, in the relation pairs with exactly one output, or 'y' value. This is the definition of a function – for every input, there is only one output.

The relation provided is: \[ \{(2, 8), (12, 3), (c, 4), (-1, 8), (0, 3)\} \]

Looking at the x-values (the first number in each ordered pair), we see the following distinct x-values: \( 2, 12, c, -1, \) and \( 0 \).

The x-values that are already provided (2, 12, -1, 0) are unique and do not repeat, meaning that their corresponding y-values (8, 3, 8, 3, respectively) are unique to them within the relation.

To make sure our relation is a function, the unknown \( c \) cannot be equal to any of these existing x-values. Therefore, \( c \) can be any real number except \( 2, 12, -1, \) and \( 0 \), since these x-values are already paired with a unique y-value.

Any other value for \( c \) will keep the relation a function, because it will not introduce a repeated x-value with a different y-value, which would violate the definition of a function.