To find the range of the relation [tex]\( 12x + 6y = 24 \)[/tex] for the domain [tex]\(\{-4, 0, 5\}\)[/tex], we need to solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
1. Start with the given equation:
[tex]\[ 12x + 6y = 24 \][/tex]
2. Isolate [tex]\( y \)[/tex]:
[tex]\[ 6y = 24 - 12x \][/tex]
[tex]\[ y = \frac{24 - 12x}{6} \][/tex]
3. Simplify the expression for [tex]\( y \)[/tex]:
[tex]\[ y = 4 - 2x \][/tex]
4. Now, we will substitute each value from the domain [tex]\(\{-4, 0, 5\}\)[/tex] into the equation [tex]\( y = 4 - 2x \)[/tex] to find the corresponding [tex]\( y \)[/tex] values (the range):
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ y = 4 - 2(-4) \][/tex]
[tex]\[ y = 4 + 8 \][/tex]
[tex]\[ y = 12 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4 - 2(0) \][/tex]
[tex]\[ y = 4 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = 4 - 2(5) \][/tex]
[tex]\[ y = 4 - 10 \][/tex]
[tex]\[ y = -6 \][/tex]
5. The corresponding range for the domain [tex]\(\{-4, 0, 5\}\)[/tex] is [tex]\(\{12, 4, -6\}\)[/tex].
The correct answer is:
A. [tex]\(\{12, 4, -6\}\)[/tex]
