In mathematics, a **surjection** is a type of function with the property that *all possible output values* of the function are generated as values of the function as the input to the function ranges over all possible input values.

More formally, a function `f`: `X` → `Y` is called **surjective** or **onto** or a **surjection** if for every `y` in the codomain `Y` there is at least one `x` in the domain `X` with `f`(`x`) = `y`.
Put another way, the range `f`(`X`) is equal to the codomain `Y`.

Surjective, not injective |
Injective, not surjective |

Bijective |
Not surjective, not injective |

When `X` and `Y` are both the real line **R**, then a surjective function `f`: **R** → **R** can be visualized as one whose graph will be intersected by *any* horizontal line.

### Examples and counterexamples

Consider the function `f`: **R** → **R** defined by `f`(`x`) = 2`x` + 1.
This function is surjective, since given an arbitrary real number `y`, we can solve `y` = 2`x` + 1 for `x` to get a solution `x` = (`y` − 1)/2.

On the other hand, the function `g`: **R** → **R** defined by `g`(`x`) = `x`^{2} is *not* surjective, because (for example) there is no real number `x` such that `x`^{2} = -1.

However, if we define the function `h`: **R** → **R**^{+} by the same formula as `g`, but with the codomain has been restricted to only the *nonnegative* real numbers, then the function `h` *is* surjective.
This is because, given an arbitrary nonnegative real number `y`, we can solve `y` = `x`^{2} to get solutions `x` = √`y` and `x` = −√`y`.

### Properties

- A function
`f`:`X`→`Y`is surjective if and only if there exists a function`g`:`Y`→`X`such that`f`^{o}`g`equals the identity function on`Y`. (This statement is equivalent to the axiom of choice.) - A function is bijective if and only if it is both surjective and injective.
- If
`f`^{o}`g`is surjective, then`f`is surjective. - If
`f`and`g`are both surjective, then`f`^{o}`g`is surjective. -
`f`:`X`→`Y`is surjective if and only if, given any functions`g`,`h`:`Y`→`Z`, whenever`g`^{o}`f`=`h`^{o}`f`, then`g`=`h`. In other words, surjective functions are precisely the epimorphisms in the category of sets. - If
`f`:`X`→`Y`is surjective and`B`is a subset of`Y`, then`f`(`f`^{ −1}(`B`)) =`B`. Thus,`B`can be recovered from its preimage`f`^{ −1}(`B`). - Every function
`h`:`X`→`Z`can be decomposed as`h`=`g`^{o}`f`for a suitable surjection`f`and injection`g`. This decomposition is unique up to isomorphism, and`f`may be thought of as a function with the same values as`h`but with its codomain restricted to the range`h`(`W`) of`h`, which is only a subset of the codomain`Z`of`h`. - If
`f`:`X`→`Y`is a surjective function, then`X`has at least as many elements as`Y`, in the sense of cardinal numbers. (This statement is also equivalent to the axiom of choice.)

See also: Injective function, Bijection