To find the range of the relation [tex]\( A: x + y = 3 \)[/tex] given the domain [tex]\((2, 1, 5)\)[/tex], follow these steps:
1. Understand the relation [tex]\( x + y = 3 \)[/tex].
2. For each value of [tex]\( x \)[/tex] from the domain, find the corresponding [tex]\( y \)[/tex] that satisfies the equation [tex]\( x + y = 3 \)[/tex].
Let's do this step by step for each [tex]\( x \)[/tex] in the given domain:
1. For [tex]\( x = 2 \)[/tex]:
- Substitute [tex]\( x = 2 \)[/tex] into the equation [tex]\( x + y = 3 \)[/tex]:
[tex]\[
2 + y = 3
\][/tex]
- Solve for [tex]\( y \)[/tex] by subtracting 2 from both sides:
[tex]\[
y = 3 - 2 = 1
\][/tex]
- So, when [tex]\( x = 2 \)[/tex], [tex]\( y = 1 \)[/tex].
2. For [tex]\( x = 1 \)[/tex]:
- Substitute [tex]\( x = 1 \)[/tex] into the equation [tex]\( x + y = 3 \)[/tex]:
[tex]\[
1 + y = 3
\][/tex]
- Solve for [tex]\( y \)[/tex] by subtracting 1 from both sides:
[tex]\[
y = 3 - 1 = 2
\][/tex]
- So, when [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex].
3. For [tex]\( x = 5 \)[/tex]:
- Substitute [tex]\( x = 5 \)[/tex] into the equation [tex]\( x + y = 3 \)[/tex]:
[tex]\[
5 + y = 3
\][/tex]
- Solve for [tex]\( y \)[/tex] by subtracting 5 from both sides:
[tex]\[
y = 3 - 5 = -2
\][/tex]
- So, when [tex]\( x = 5 \)[/tex], [tex]\( y = -2 \)[/tex].
Based on these calculations, the corresponding [tex]\( y \)[/tex]-values, which form the range, are [tex]\((1, 2, -2)\)[/tex]. Therefore, the range for the relation [tex]\( A \)[/tex] given the domain [tex]\((2, 1, 5)\)[/tex] is:
[tex]\[ \boxed{(1, 2, -2)} \][/tex]
None of the provided multiple choice answers match this range explicitly.
