One-to-one functions never send two different inputs to the same output
A function is injective when no two distinct inputs ever land on the same output. That simple rule lets you undo the function from the left, powers a quick graph test with a horizontal ruler, and quietly underpins how mathematicians compare the sizes of sets, infinite ones included.
Formally, a function f is injective, also called an injection or one-to-one, when f(a) = f(b) forces a = b. The contrapositive says the same thing another way: different inputs yield different outputs. So each element of the target set is hit by at most one element of the domain. A function that fails this is sometimes called many-to-one. The phrase one-to-one should not be mixed up with one-to-one correspondence, which means a bijection, where every target element is hit exactly once.
Some cases are automatic. The identity function on any set is injective and in fact bijective. A function from the empty set, the empty function, qualifies trivially, and so does any function whose domain contains a single element. The real exponential function is injective but not surjective, since no real input produces a negative result. For functions from the reals to the reals there is a visual check: draw horizontal lines across the graph, and if none meets it more than once, the function is one-to-one.
Injectivity is closely tied to reversibility. Any function with a left inverse, some g with g(f(x)) = x, must be injective, and any injection with a non-empty domain has one. That does not make it fully invertible, which needs a bijection, but shrinking the target to the actual range turns an injection into a bijection. Injective partial functions are known as partial bijections.
In algebra, a structure-preserving map that is injective is usually called a monomorphism, and for common structures such as vector spaces the two notions coincide as a theorem, though category theory defines monomorphisms more generally. In the category of sets, injections are exactly the monomorphisms. Writers often mark injections with a tailed or hooked arrow, and some reserve the hooked version for inclusion maps.
Source: Injective function
