To find the range for the given relation \( 12x + 6y = 24 \) when the domain is \( \{-4, 0, 5\} \), we need to solve for \( y \) in terms of \( x \).
Given the equation:
[tex]\[ 12x + 6y = 24 \][/tex]
First, isolate \( y \):
[tex]\[ 6y = 24 - 12x \][/tex]
[tex]\[ y = \frac{24 - 12x}{6} \][/tex]
[tex]\[ y = 4 - 2x \][/tex]
Now, we will substitute each value of the domain into this equation to find the corresponding \( y \)-values, which will give us the range.
1. When \( x = -4 \):
[tex]\[ y = 4 - 2(-4) \][/tex]
[tex]\[ y = 4 + 8 \][/tex]
[tex]\[ y = 12 \][/tex]
2. When \( x = 0 \):
[tex]\[ y = 4 - 2(0) \][/tex]
[tex]\[ y = 4 \][/tex]
3. When \( x = 5 \):
[tex]\[ y = 4 - 2(5) \][/tex]
[tex]\[ y = 4 - 10 \][/tex]
[tex]\[ y = -6 \][/tex]
So, the \( y \)-values (the range) corresponding to the domain \(\{-4, 0, 5\}\) are:
[tex]\[ \{12, 4, -6\} \][/tex]
Thus, the correct answer is:
D. [tex]\( \{12, 4, -6\} \)[/tex]
