# Surjective function facts for kids

Surjection. There is an arrow to every element in the codomain B from (at least) one element of the domain A. |

In mathematics, a **surjective** or **onto** function is a function *f* : *A* → *B* with the following property. For every element *b* in the codomain *B*, there is at least one element *a* in the domain *A* such that *f*(*a*)=*b*. This means that no element in the codomain is unmapped, and that the range and codomain of *f* are the same set.

The term *surjection* and the related terms *injection* and *bijection* were introduced by the group of mathematicians that called itself . In the 1930s, this group of mathematicians published a series of books on modern advanced mathematics. The French prefix *sur* means *above* or *onto* and was chosen since a surjective function maps its domain **on to** its codomain.

Not a surjection. No element in the domain A is mapped to the element {4} in the codomain B. |

## Contents

## Basic properties

Formally:

- is a surjective function if such that

where the element is called the **image** of the element , and the element a **pre-image** of the element .

The formal definition can also be interpreted in two ways:

- Every element of the codomain
*B*is the*image*of at least one element in the domain*A*. - Every element of the codomain
*B*has at least one pre-image in the domain*A*.

A pre-image does *not* have to be unique. In the top image, both {X} and {Y} are pre-images of the element {1}. It is only important that there be at least one pre-image.

## Examples

### Elementary functions

Let *f*(*x*):ℝ→ℝ be a real-valued function *y*=*f*(*x*) of a real-valued argument *x*. (This means both the input and output are numbers.)

**Graphic meaning**: The function*f*is a surjection if every horizontal line intersects the graph of*f*in at least one point.**Analytic meaning**: The function*f*is a surjection if for every real number*y*_{o}we can find at least one real number*x*_{o}such that*y*_{o}=*f*(*x*_{o}).

Finding a pre-image *x*_{o} for a given *y*_{o} is equivalent to either question:

- Does the equation
*f*(*x*)-*y*_{o}=0 have a solution? or - Does the function
*f*(*x*)-*y*_{o}have a root?

In mathematics, we can find exact (analytic) roots only of polynomials of first, second (and third) degree. We find roots of all other functions approximately (numerically). This means a formal proof of surjectivity is rarely direct. So the discussions below are informal.

**Example:** The linear function of a slanted line is *onto*. That is, *y*=*ax*+*b* where *a*≠0 is a surjection. (It is also an injection and thus a bijection.)

- Proof: Substitute
*y*_{o}into the function and solve for*x*. Since*a*≠0 we get*x*= (*y*_{o}-*b*)/_{a}. This means*x*_{o}=(*y*_{o}-*b*)/_{a}is a pre-image of*y*_{o}. This proves that the function*y*=*ax*+*b*where*a*≠0 is a surjection. (Since there is exactly one pre-image, this function is also an injection.) - Practical example:
*y*= –2*x*+4. What is the pre-image of*y*=2? Solution: Here*a*= –2, i.e.*a*≠0 and the question is: For what*x*is*y*=2? We substitute*y*=2 into the function. We get*x*=1, i.e.*y*(1)=2. So the answer is:*x*=1 is the pre-image of*y*=2.

**Example:** The cubic polynomial (of third degree) *f*(*x*)=*x*^{3}-3*x* is a surjection.

- Discussion: The cubic equation
*x*^{3}-3*x*-*y*_{o}=0 has real coefficients (*a*_{3}=1,*a*_{2}=0,*a*_{1}=–3,*a*_{0}=–*y*_{o}). Every such cubic equation has at least one real root. Since the domain of the polynomial is ℝ, the means that ther is at least one pre-image*x*_{o}in the domain. That is, (*x*_{0})^{3}-3*x*_{0}-*y*_{o}=0. So the function is a surjection. (However, this function is not an injection. For example,*y*_{o}=2 has 2 pre-images:*x*=–1 and*x*=2. In fact, every*y*, –2≤*y*≤2 has at least 2 pre-images.)

**Example:** The quadratic function *f*(*x*) = *x*^{2} is **not** a surjection. There is no *x* such that *x*^{2} = −1. The range of *x*² is [0,+∞) , that is, the set of non-negative numbers. (Also, this function is not an injection.)

Note: One can make a non-surjective function into a surjection by *restricting* its codomain to elements of its range. For example, the *new* function, *f*_{N}(*x*):ℝ → [0,+∞) where *f*_{N}(*x*) = *x*^{2} is a surjective function. (This is not the same as the restriction of a function which restricts the domain!)

**Example:** The exponential function *f*(*x*) = 10^{x} is **not** a surjection. The range of *10*^{x} is (0,+∞), that is, the set of positive numbers. (This function is an injection.)

### Other examples with real-valued functions

**Example:** The **logarithmic function base 10** *f*(*x*):(0,+∞)→ℝ defined by *f*(*x*)=log(*x*) or *y*=log_{10}(*x*) is a surjection (and an injection). (This is the inverse function of 10^{x}.)

- The projection of a Cartesian product
*A*×*B*onto one of its factors is a surjection.

**Example:** The function *f*((*x*,*y*)):ℝ²→ℝ defined by *z*=*y* is a surjection. Its graph is a plane in 3-dimensional space. The pre-image of *z*_{o} is the line y=*z*_{o} in the *x*0*y* plane.

- In 3D games, 3-dimensional space is projected onto a 2-dimensional screen with a surjection.

## Related pages

interactive quiz interactive

## See also

In Spanish: Función sobreyectiva para niños

Delilah Pierce |

Gordon Parks |

Augusta Savage |

Charles Ethan Porter |

*Kiddle Encyclopedia.*