If R={(1,1),(2,2),(2,3)}, why is R not a function?

Correct answer: Because 2 has two images

The governing concept is the uniqueness condition for a function. A relation from a domain to a codomain is a function only if every input, or first component, is associated with exactly one output, or image. In R, the input 2 occurs in both (2, 2) and (2, 3), so the same input has two different images, 2 and 3. This violates the definition of a function, making option A correct. The pair (1, 1) is acceptable because input 1 has only one image. Having 3 as a second component is not a problem, and the fact that the relation has three pairs does not by itself decide whether it is a function. The failure is specifically the repeated input with different outputs.