Injectively

Injectively is an adverb used in mathematics to describe how a function behaves with respect to distinct inputs. When a function acts injectively, it maps each element of its domain to a unique element of its codomain, so that no two different inputs share the same output. The term describes the manner of the mapping, not the object itself.

Formally, a function f: X → Y is injective if, whenever f(x1) = f(x2), it follows that x1 = x2. Equivalently, f(x1) ≠ f(x2) for all x1 ≠ x2. In this language, one can say that f is injective, or that f maps X injectively into Y. The closure of this idea is that the preimage of any single y in the image has at most one element.

Common ways to recognize injectivity include the horizontal line test for real-valued functions on the real line, or the existence of a left inverse g with g∘f = id on the domain. Differentiable functions that are strictly monotone on an interval are injective on that interval. Conversely, a function like f(x) = x^2 is not injective on the real numbers, though it is injective on [0, ∞).

Injectivity is one of the basic properties alongside surjectivity and bijectivity. Understanding injectivity helps in analyzing mappings, solving equations, and determining whether inverses exist on restricted domains. An adjective form is injective, while the adverb injectively is used to describe the manner in which the mapping operates.