Surjective function

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.^[1][2][3]

The term surjection and the related terms injection and bijection were introduced by the group of mathematicians that called itself Nicholas Bourbaki.^[4] 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.

Formally:^[5]

${\displaystyle f:A\rightarrow B}$  is a surjective function if  ${\displaystyle \forall b\in B\,\,\exists a\in A}$ such that ${\displaystyle f(a)=b\,.}$

where the element ${\displaystyle b}$ is called the image of the element ${\displaystyle a}$, and the element ${\displaystyle a}$ a pre-image of the element ${\displaystyle b}$.

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.

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= –2x+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)=x³-3x is a surjection.

Discussion: The cubic equation x³-3x-y_o=0 has real coefficients (a₃=1, a₂=0, a₁=–3, a₀=–y_o). Every such cubic equation has at least one real root.^[6] 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, (x₀)³-3x₀-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) = x² is not a surjection. There is no x such that x² = −1. The range of x² 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) = x² 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.)

Surjection. f(x):ℝ→ℝ (and injection)	Surjection. f(x):ℝ→ℝ (not an injection)	Not a surjection. f(x):ℝ→ℝ (nor an injection)
Not a surjection. f(x):ℝ→ℝ (but is an injection)	Surjection. f(x):(0,+∞)→ℝ (and injection)	Surjection. z:ℝ²→ℝ, z=y. (The picture shows that the pre-image of z=2 is the line y=2.)

Example: The logarithmic function base 10 f(x):(0,+∞)→ℝ defined by f(x)=log(x) or y=log₁₀(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 x0y plane.

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

• Injective function
• Bijective function

• "Injective, Surjective, Bijective". 2013. Retrieved 2013-12-01. interactive quiz
• "Injectivity, Surjectivity". Wolfram alpha. Archived from the original on 2013-11-25. Retrieved 2013-12-01. interactive