To find the range for the given relation [tex]\( 3x + y = 3 \)[/tex] with the domain [tex]\( \{-2, 2, 4\} \)[/tex], we need to determine the corresponding [tex]\( y \)[/tex]-values for each given [tex]\( x \)[/tex]-value in the domain. Here is the step-by-step process:
1. Substitute [tex]\( x = -2 \)[/tex] into the equation [tex]\( 3x + y = 3 \)[/tex]:
[tex]\[
3(-2) + y = 3
\][/tex]
This simplifies to:
[tex]\[
-6 + y = 3
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y = 3 + 6
\][/tex]
[tex]\[
y = 9
\][/tex]
2. Substitute [tex]\( x = 2 \)[/tex] into the equation [tex]\( 3x + y = 3 \)[/tex]:
[tex]\[
3(2) + y = 3
\][/tex]
This simplifies to:
[tex]\[
6 + y = 3
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y = 3 - 6
\][/tex]
[tex]\[
y = -3
\][/tex]
3. Substitute [tex]\( x = 4 \)[/tex] into the equation [tex]\( 3x + y = 3 \)[/tex]:
[tex]\[
3(4) + y = 3
\][/tex]
This simplifies to:
[tex]\[
12 + y = 3
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y = 3 - 12
\][/tex]
[tex]\[
y = -9
\][/tex]
Thus, the range corresponding to the domain [tex]\( \{-2, 2, 4\} \)[/tex] for the relation [tex]\( 3x + y = 3 \)[/tex] is [tex]\( \{9, -3, -9\} \)[/tex].
Therefore, the correct answer is:
A. [tex]\( \{9, -3, -9\} \)[/tex]
