The relation R = {(x,y): y² = x − 1, x ∈ {2,5,10}, y ∈ {−3,−2,−1,1,2,3}} is considered from X = {2,5,10} to Y = {−3,−2,−1,1,2,3}. What is the correct conclusion?

Correct answer: It is not a function because 2 has two images

To test whether a relation from X to Y is a function, every element of the domain X must have exactly one image in Y. Use y² = x − 1. For x = 2, y² = 1, so y = 1 or y = −1; both ordered pairs (2,1) and (2,−1) belong to R. Thus the single input 2 has two images, which violates the defining uniqueness condition for a function. For completeness, x = 5 gives y = 2 or −2, and x = 10 gives y = 3 or −3, so each input actually has two images. Therefore option B states a valid and sufficient reason. Option C is false because 10 does have images, while A and D incorrectly call the relation a function. The relation is many-valued, not a function.