To find the range of the relation [tex]\( 12x + 6y = 24 \)[/tex] given the domain [tex]\(\{-4, 0, 5\}\)[/tex], we need to find the corresponding [tex]\( y \)[/tex] values for each [tex]\( x \)[/tex] in the domain.
The relation is:
[tex]\[12x + 6y = 24\][/tex]
First, solve for [tex]\( y \)[/tex]:
[tex]\[12x + 6y = 24 \rightarrow 6y = 24 - 12x \rightarrow y = \frac{24 - 12x}{6} \rightarrow y = 4 - 2x\][/tex]
Now, we will calculate [tex]\( y \)[/tex] for each [tex]\( x \)[/tex] in the domain [tex]\(\{-4, 0, 5\}\)[/tex]:
1. For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = 4 - 2(-4) \][/tex]
[tex]\[ y = 4 + 8 \][/tex]
[tex]\[ y = 12 \][/tex]
2. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4 - 2(0) \][/tex]
[tex]\[ y = 4 \][/tex]
3. For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = 4 - 2(5) \][/tex]
[tex]\[ y = 4 - 10 \][/tex]
[tex]\[ y = -6 \][/tex]
Thus, the range for the given domain is [tex]\(\{12, 4, -6\}\)[/tex].
Therefore, the correct answer is:
A. [tex]\(\{12, 4, -6\}\)[/tex]
