The relation \(R=\{(x,y): |x+y|=2,\ x\in\{-2,0,2\},\ y\in\{-4,-2,0,2,4\}\}\) is considered from the domain \(\{-2,0,2\}\) to the codomain \(\{-4,-2,0,2,4\}\). Why is this relation not a function?

Correct answer: Because \(x=0\) has two images, \(y=2\) and \(y=-2\)

Putting \(x=0\) into \(|x+y|=2\) gives \(|y|=2\), so both \(y=2\) and \(y=-2\) form ordered pairs in the relation. A function must assign exactly one image to each domain element; two images for the same \(x\) violate this condition. In fact, \(x=2\) gives \(y=0,-4\), and \(x=-2\) gives \(y=0,4\), so options B and C are also incorrect. Exam tip: to test a relation for being a function, check that every domain element has exactly one image.