Injective function

In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its codomain.[1] In other words, every element of the function's codomain is the image of at most one element of its domain.[2] The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an injective homomorphism is also called a monomorphism. However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism.[3] This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism for more details.

A function f that is not injective is sometimes called many-to-one.[2]

More generally, when X and Y are both the real line R, then an injective function f : RR is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the horizontal line test.[2]

Functions with left inverses are always injections. That is, given f : XY, if there is a function g : YX such that for every xX,

Conversely, every injection f with non-empty domain has a left inverse g, which can be defined by fixing an element a in the domain of f so that g(x) equals the unique pre-image of x under f if it exists and g(x) = a otherwise.[5]

The left inverse g is not necessarily an inverse of f, because the composition in the other order, fg, may differ from the identity on Y. In other words, an injective function can be "reversed" by a left inverse, but is not necessarily invertible, which requires that the function is bijective.

In fact, to turn an injective function f : XY into a bijective (hence invertible) function, it suffices to replace its codomain Y by its actual range J = f(X). That is, let g : XJ such that g(x) = f(x) for all x in X; then g is bijective. Indeed, f can be factored as inclJ,Yg, where inclJ,Y is the inclusion function from J into Y.

More generally, injective partial functions are called partial bijections.

A proof that a function f is injective depends on how the function is presented and what properties the function holds. For functions that are given by some formula there is a basic idea. We use the definition of injectivity, namely that if f(x) = f(y), then x = y.[6]

Proof: Let f : XY. Suppose f(x) = f(y). So 2x + 3 = 2y + 32x = 2yx = y. Therefore, it follows from the definition that f is injective.

There are multiple other methods of proving that a function is injective. For example, in calculus if f is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval. In linear algebra, if f is a linear transformation it is sufficient to show that the kernel of f contains only the zero vector. If f is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list.

A graphical approach for a real-valued function f of a real variable x is the horizontal line test. If every horizontal line intersects the curve of f(x) in at most one point, then f is injective or one-to-one.