A mathematical function is called

**injective**(or

**one-to-one**or an

**injection**) if the function maps no more than one possible input value to each possible output value. (This is in contrast to a "many to one" function, which maps two or more input values to some output values).

More formally, a function `f`: `X` → `Y` is injective if for every `y` in the codomain `Y` there is at most one `x` in the domain `X` with `f`(`x`) = `y`.
Put another way, given `x` and `x`' in `X`, if `f`(`x`) = `f`(`x`'), then it follows that `x` = `x`'.

Surjective, not injective |
Injective, not surjective |

Bijective |
Not surjective, not injective |

When `X` and `Y` are both the real line **R**, then an injective function `f`: **R** → **R** can be visualized as one whose graph is never intersected by any horizontal line more than once.
(This is the *horizontal line test*.)

### Examples and counterexamples

Consider the function `f`: **R** → **R** defined by `f`(`x`) = 2`x` + 1.
This function is injective, since given arbitrary real numbers `x` and `x`', if 2`x` + 1 = 2`x`' + 1, then 2`x` = 2`x`', so `x` = `x`'.

On the other hand, the function `g`: **R** → **R** defined by `g`(`x`) = `x`^{2} is *not* injective, because (for example) `g`(1) = 1 = `g`(−1).

However, if we define the function `h`: **R**^{+} → **R** by the same formula as `g`, but with the domain restricted to only the *nonnegative* real numbers, then the function `h` *is* injective.
This is because, given arbitrary nonnegative real numbers `x` and `x`', if `x`^{2} = `x`'^{2}, then |`x`| = |`x`'|, so `x` = `x`'.

### Properties

- A function
`f`:`X`→`Y`is injective if and only if`X`is the empty set or there exists a function`g`:`Y`→`X`such that`g`o`f`equals the identity function on`X`. - A function is bijective if and only if it is both injective and surjective.
- If
`g`o`f`is injective, then`f`is injective. - If
`f`and`g`are both injective, then`g`o`f`is injective. -
`f`:`X`→`Y`is injective if and only if, given any functions`g`,`h`:`W`→`X`, whenever`f`o`g`=`f`o`h`, then`g`=`h`. In other words, injective functions are precisely the monomorphisms in the category of sets. - If
`f`:`X`→`Y`is injective and`A`is a subset of`X`, then`f`^{ −1}(`f`(`A`)) =`A`. Thus,`A`can be recovered from its image`f`(`A`). - If
`f`:`X`→`Y`is injective and`A`and`B`are both subsets of`X`, then`f`(`A`∩`B`) =`f`(`A`) ∩`f`(`B`). - Every function
`h`:`W`→`Y`can be decomposed as`h`=`f`o`g`for a suitable injection`f`and surjection`g`. This decomposition is unique up to isomorphism, and`f`may be thought of as the inclusion function of the range`h`(`W`) of`h`as a subset of the codomain`Y`of`h`. - If
`f`:`X`→`Y`is an injective function, then`Y`has at least as many elements as`X`, in the sense of cardinal numbers.

See also: Surjection, Bijection