Noninjectivity

Noninjectivity refers to the property of a function failing to be injective. A function f: A → B is injective if distinct inputs produce distinct outputs; that is, f(x1) = f(x2) implies x1 = x2 for all x1, x2 in A. A function is noninjective if this implication fails for some pair of distinct inputs, meaning there exist x1 ≠ x2 with f(x1) = f(x2).

Examples: The real function f(x) = x^2 from R to R is noninjective because f(1) = f(-1) = 1. The constant function f(x) = c is noninjective on any nonempty domain. A function can be noninjective yet surjective or neither.

Tests and related concepts: For real-valued functions, the horizontal line test detects noninjectivity: a horizontal line intersects the graph in more than one point shows noninjectivity. For linear transformations between vector spaces, noninjectivity is equivalent to having a nontrivial kernel (the set of vectors mapped to zero is not only the zero vector).

Consequences and remedies: Noninjective maps do not have two-sided inverses, and cannot be reversed uniquely on their image. One can obtain an injective map by restricting the domain or codomain, or by passing to equivalence classes where inputs that share the same image are identified (the quotient by the kernel).

See also: Injective function, Surjective function, Bijective, Kernel, Horizontal line test.