In mathematics, an injective function (also known as injection, or one-to-one function^[1]) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x₁ ≠ x₂ implies f(x₁) ≠ f(x₂) (equivalently by contraposition, f(x₁) = f(x₂) implies x₁ = x₂). 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 👁 {\displaystyle f}
that is not injective is sometimes called many-to-one.^[2]

Let 👁 {\displaystyle f}
be a function whose domain is a set ⁠👁 {\displaystyle X}
⁠. The function 👁 {\displaystyle f}
is said to be injective provided that for all 👁 {\displaystyle a}
and 👁 {\displaystyle b}
in 👁 {\displaystyle X,}
if ⁠👁 {\displaystyle f(a)=f(b)}
⁠, then ⁠👁 {\displaystyle a=b}
⁠; that is, 👁 {\displaystyle f(a)=f(b)}
implies ⁠👁 {\displaystyle a=b}
⁠. Equivalently, if ⁠👁 {\displaystyle a\neq b}
⁠, then 👁 {\displaystyle f(a)\neq f(b)}
in the contrapositive statement.

Symbolically,👁 {\displaystyle \forall a,b\in X,\;\;f(a)=f(b)\Rightarrow a=b,}
which is logically equivalent to the contrapositive,^[4]👁 {\displaystyle \forall a,b\in X,\;\;a\neq b\Rightarrow f(a)\neq f(b).}
An injective function (or, more generally, a monomorphism) is often denoted by using the specialized arrows ↣ or ↪ (for example, 👁 {\displaystyle f:A\rightarrowtail B}
or ⁠👁 {\displaystyle f:A\hookrightarrow B}
⁠), although some authors specifically reserve ↪ for an inclusion map.^[5]

For visual examples, readers are directed to the gallery section.

• For any set 👁 {\displaystyle X}
  and any subset ⁠👁 {\displaystyle S\subseteq X}
  ⁠, the inclusion map 👁 {\displaystyle S\to X}
  (which sends any element 👁 {\displaystyle s\in S}
  to itself) is injective. In particular, the identity function 👁 {\displaystyle X\to X}
  is always injective (and in fact bijective).
• If the domain of a function is the empty set, then the function is the empty function, which is injective.
• If the domain of a function has one element (that is, it is a singleton set), then the function is always injective.
• The function 👁 {\displaystyle f:\mathbb {R} \to \mathbb {R} }
  defined by 👁 {\displaystyle f(x)=2x+1}
  is injective.
• The function 👁 {\displaystyle g:\mathbb {R} \to \mathbb {R} }
  defined by 👁 {\displaystyle g(x)=x^{2}}
  is not injective, because (for example) 👁 {\displaystyle g(1)=1=g(-1).}
  However, if 👁 {\displaystyle g}
  is redefined so that its domain is the non-negative real numbers [0, +∞), then 👁 {\displaystyle g}
  is injective.
• The exponential function 👁 {\displaystyle \exp :\mathbb {R} \to \mathbb {R} }
  defined by 👁 {\displaystyle \exp(x)=e^{x}}
  is injective (but not surjective, as no real value maps to a negative number).
• The natural logarithm function 👁 {\displaystyle \ln :(0,\infty )\to \mathbb {R} }
  defined by 👁 {\displaystyle x\mapsto \ln x}
  is injective.
• The function 👁 {\displaystyle g:\mathbb {R} \to \mathbb {R} }
  defined by 👁 {\displaystyle g(x)=x^{n}-x}
  is not injective, since, for example, ⁠👁 {\displaystyle g(0)=g(1)=0}
  ⁠.

More generally, when 👁 {\displaystyle X}
and 👁 {\displaystyle Y}
are both the real line ⁠👁 {\displaystyle \mathbb {R} }
⁠, then an injective function 👁 {\displaystyle f:\mathbb {R} \to \mathbb {R} }
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 ⁠👁 {\displaystyle f:X\to Y}
⁠, if there is a function 👁 {\displaystyle g:Y\to X}
such that for every ⁠👁 {\displaystyle x\in X}
⁠, ⁠👁 {\displaystyle g(f(x))=x}
⁠, then 👁 {\displaystyle f}
is injective. The proof is that 👁 {\displaystyle f(a)=f(b)\rightarrow g(f(a))=g(f(b))\rightarrow a=b.}

In this case, 👁 {\displaystyle g}
is called a retraction of ⁠👁 {\displaystyle f}
⁠. Conversely, 👁 {\displaystyle f}
is called a section of ⁠👁 {\displaystyle g}
⁠. For example: 👁 {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} ^{2},x\mapsto (1,m)^{\intercal }x}
is retracted by ⁠👁 {\displaystyle g:y\mapsto {\frac {(1,m)}{1+m^{2}}}y}
⁠.

Conversely, every injection 👁 {\displaystyle f}
with a non-empty domain has a left inverse 👁 {\displaystyle g}
. It can be defined by choosing an element 👁 {\displaystyle a}
in the domain of 👁 {\displaystyle f}
and setting 👁 {\displaystyle g(y)}
to the unique element of the pre-image 👁 {\displaystyle f^{-1}[y]}
(if it is non-empty) or to 👁 {\displaystyle a}
(otherwise).^[6]

The left inverse 👁 {\displaystyle g}
is not necessarily an inverse of 👁 {\displaystyle f,}
because the composition in the other order, ⁠👁 {\displaystyle f\circ g}
⁠, may differ from the identity on ⁠👁 {\displaystyle 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 👁 {\displaystyle f:X\to Y}
into a bijective (hence invertible) function, it suffices to replace its codomain 👁 {\displaystyle Y}
by its actual image 👁 {\displaystyle J=f(X).}
That is, let 👁 {\displaystyle g:X\to J}
such that 👁 {\displaystyle g(x)=f(x)}
for all ⁠👁 {\displaystyle x\in X}
⁠; then 👁 {\displaystyle g}
is bijective. Indeed, 👁 {\displaystyle f}
can be factored as ⁠👁 {\displaystyle \operatorname {In} _{J,Y}\circ g}
⁠, where 👁 {\displaystyle \operatorname {In} _{J,Y}}
is the inclusion function from 👁 {\displaystyle J}
into ⁠👁 {\displaystyle Y}
⁠.

More generally, injective partial functions are called partial bijections.

• If 👁 {\displaystyle f}
  and 👁 {\displaystyle g}
  are both injective then 👁 {\displaystyle f\circ g}
  is injective.
• If 👁 {\displaystyle g\circ f}
  is injective, then 👁 {\displaystyle f}
  is injective (but 👁 {\displaystyle g}
  need not be).
• 👁 {\displaystyle f:X\to Y}
  is injective if and only if, given any functions ⁠👁 {\displaystyle g}
  ⁠, 👁 {\displaystyle h:W\to X}
  whenever ⁠👁 {\displaystyle f\circ g=f\circ h}
  ⁠, then ⁠👁 {\displaystyle g=h}
  ⁠. In other words, injective functions are precisely the monomorphisms in the category Set of sets.
• If 👁 {\displaystyle f:X\to Y}
  is injective and 👁 {\displaystyle A}
  is a subset of ⁠👁 {\displaystyle X}
  ⁠, then ⁠👁 {\displaystyle f^{-1}(f(A))=A}
  ⁠. Thus, 👁 {\displaystyle A}
  can be recovered from its image ⁠👁 {\displaystyle f(A)}
  ⁠.
• If 👁 {\displaystyle f:X\to Y}
  is injective and 👁 {\displaystyle A}
  and 👁 {\displaystyle B}
  are both subsets of ⁠👁 {\displaystyle X}
  ⁠, then ⁠👁 {\displaystyle f(A\cap B)=f(A)\cap f(B)}
  ⁠.
• Every function 👁 {\displaystyle h:W\to Y}
  can be decomposed as 👁 {\displaystyle h=f\circ g}
  for a suitable injection 👁 {\displaystyle f}
  and surjection ⁠👁 {\displaystyle g}
  ⁠. This decomposition is unique up to isomorphism, and 👁 {\displaystyle f}
  may be thought of as the inclusion function of the range 👁 {\displaystyle h(W)}
  of 👁 {\displaystyle h}
  as a subset of the codomain 👁 {\displaystyle Y}
  of ⁠👁 {\displaystyle h}
  ⁠.
• If 👁 {\displaystyle f:X\to Y}
  is an injective function, then 👁 {\displaystyle Y}
  has at least as many elements as 👁 {\displaystyle X,}
  in the sense of cardinal numbers. In particular, if, in addition, there is an injection from ⁠👁 {\displaystyle Y}
  ⁠ to ⁠👁 {\displaystyle X}
  ⁠, then 👁 {\displaystyle X}
  and 👁 {\displaystyle Y}
  have the same cardinal number. (This is known as the Cantor–Bernstein–Schroeder theorem.)
• If both 👁 {\displaystyle X}
  and 👁 {\displaystyle Y}
  are finite with the same number of elements, then 👁 {\displaystyle f:X\to Y}
  is injective if and only if 👁 {\displaystyle f}
  is surjective (in which case 👁 {\displaystyle f}
  is bijective).
• An injective function which is a homomorphism between two algebraic structures is an embedding.
• Unlike surjectivity, which is a relation between the graph of a function and its codomain, injectivity is a property of the graph of the function alone; that is, whether a function 👁 {\displaystyle f}
  is injective can be decided by only considering the graph (and not the codomain) of ⁠👁 {\displaystyle f}
  ⁠.

A proof that a function 👁 {\displaystyle 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 ⁠👁 {\displaystyle f(x)=f(y)}
⁠, then ⁠👁 {\displaystyle x=y}
⁠.^[7]

Here is an example: 👁 {\displaystyle f(x)=2x+3}

Proof: Let ⁠👁 {\displaystyle f:X\to Y}
⁠. Suppose ⁠👁 {\displaystyle f(x)=f(y)}
⁠. So 👁 {\displaystyle 2x+3=2y+3}
implies ⁠👁 {\displaystyle 2x=2y}
⁠, which implies ⁠👁 {\displaystyle x=y}
⁠. Therefore, it follows from the definition that 👁 {\displaystyle f}
is injective.

There are multiple other methods of proving that a function is injective. For example, in calculus if 👁 {\displaystyle 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 👁 {\displaystyle f}
is a linear transformation it is sufficient to show that the kernel of 👁 {\displaystyle f}
contains only the zero vector. If 👁 {\displaystyle 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 👁 {\displaystyle f}
of a real variable 👁 {\displaystyle x}
is the horizontal line test. If every horizontal line intersects the curve of 👁 {\displaystyle f(x)}
in at most one point, then 👁 {\displaystyle f}
is injective or one-to-one.

• Bijection, injection and surjection – Properties of mathematical functions
• Injective metric space – Type of metric space
• Monotonic function – Order-preserving mathematical function
• Univalent function – Mathematical concept

• Bartle, Robert G. (1976), The Elements of Real Analysis (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-05464-1, p. 17 ff.
• Halmos, Paul R. (1974), Naive Set Theory, New York: Springer, ISBN 978-0-387-90092-6, p. 38 ff.

• Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.
• Khan Academy – Surjective (onto) and Injective (one-to-one) functions: Introduction to surjective and injective functions