If R = {(x, y) : x, y ∈ Z, x² + y² = 25}, Find Domain and Range
Question
If
\[ R = \{(x,y): x,y\in Z,\ x^2+y^2=25\} \]
then Domain \( (R) \) = ……………. and Range \( (R) \) = …………
Solution
We need to find all integer pairs \( (x,y) \) satisfying:
\[ x^2+y^2=25 \]
Since
\[ 25 = 5^2 \]
possible integer solutions are:
\[ (0,5),\ (0,-5),\ (5,0),\ (-5,0), \]
\[ (3,4),\ (3,-4),\ (-3,4),\ (-3,-4), \]
\[ (4,3),\ (4,-3),\ (-4,3),\ (-4,-3) \]
Therefore, the domain is the set of all first components:
\[ \text{Domain}(R)=\{-5,-4,-3,0,3,4,5\} \]
The range is the set of all second components:
\[ \text{Range}(R)=\{-5,-4,-3,0,3,4,5\} \]
Hence,
\[ \boxed{\text{Domain}(R)=\{-5,-4,-3,0,3,4,5\}} \]
and
\[ \boxed{\text{Range}(R)=\{-5,-4,-3,0,3,4,5\}} \]