n. a relation between two sets that associates a unique element of the second with each element of the first; a many-one relation, written f or f(x); formally, the set of ordered pairs 〈x, f(x)〉. If we write y = f(x), then y is the value of the function for the argument x; if the function is defined between sets S and T with the arguments in S and the values in T, then S is the domain and T the codomain of the function. We can write or If s is a subset of S, then f(s) is the set of values of f(x) for x ∈ s, and is called the image of s under the function. The image, f(S), of the domain is the range of the function. Although the terms are usually regarded as synonymous, some authors prefer the term mapping, or transformation when dealing with abstract spaces; some use the former terms to indicate that the identity of the function is taken to depend on the specified domain and range as well as on the set of ordered pairs of relata, so that the real-valued square root function is regarded as a different mapping when defined on all reals and when defined on the non-negative reals; `transformation' is often preferred when the algebraic expression for the value of the function is derived in a uniform way from the expression for the argument. Compare graph, set-valued mapping.