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Surjective Function

A surjective function is defined between set A and set B, such that every element of set B is associated with at least one element of set A. The domain and range of a surjective function are equal.

Let us learn more about the surjective function, along with its properties and examples.

1.	What Is a Surjective Function?
2.	Properties of Surjective Function
4.	Examples on Surjective Function
5.	Practice Questions on Surjective Function
6.	FAQs on Surjective Function

What Is a Surjective Function?

Surjective function is defined with reference to the elements of the range set, such that every element of the range is a co-domain. A surjective function is a function whose image is equal to its co-domain. Also, the range, co-domain and the image of a surjective function are all equal. Additionally, we can say that a subjective function is an onto function when every y ∈ co-domain has at least one pre-image x ∈ domain such that f(x) = y. Let's go ahead and explore more about surjective function.

A function 'f' from set A to set B is called a surjective function if for each b ∈ B there exists at least one a ∈ A such that f(a) = b. None of the elements are left out in the onto function because they are all mapped from some element of set A. Consider the example given below:

Let A = {a₁, a₂, a₃ } and B = {b₁, b₂ } then f : A →B.:{(a_1,b_1),(a₂, b₂), (a₃, b₂)}

Here in the above example, every element of set B has been utilized, and every element of set B is an image of one or more than one element of set A.

Surjective Function

In the above examples of functions, the functions which do not have any remaining element in set B is a surjective function. In a surjective function, every element of set B has been mapped from one or more than one element of set A. Also, the functions which are not surjective functions have elements in set B that have not been mapped from any element of set A.

Properties of Surjective Function

A function is considered to be a surjective function only if the range is equal to the co-domain. Here are some of the important properties of surjective function:

In a surjective function, every element in the co-domain will be assigned to at least one element of the domain.
The co-domain element in a subjective function can be an image for more than one element of the domain set.
In a subjective function, the co-domain is equal to the range.A function f: A →B is an onto, or surjective, function if the range of f equals the co-domain of the function f.
Every function that is a surjective function has a right inverse. Also, every function which has a right inverse can be considered as a surjective function.

Related Topics

The following topics help in a better understanding of surjective function.

Examples on Surjective Function

Example 1: Given that the set A = {1, 2, 3}, set B = {4, 5} and let the function f = {(1, 4), (2, 5), (3, 5)}. Show that the function f is a surjective function from A to B.

Solution: Domain = set A = {1, 2, 3}

We can see that the element from set A,1 has an image 4, and both 2 and 3 have the same image 5. Thus, the range of the function is {4, 5} which is equal to set B. So we conclude that F: A →B is an onto function.

Therefore, the given function f is a surjective function.
Example 2: Identify, if the function f : R → R defined by g(x) = 1 + x², is a surjective function.

Solution: The given function is g(x) = 1 + x².

For the set of real numbers, we know that x² > 0. So 1 + x² > 1. g(x) > 1 and hence the range of the function is (1, ∞). Whereas, the second set is R (Real Numbers). So the range is not equal to co-domain and hence the function is not a surjective function.
Example 3: Prove if the function g : R → R defined by g(x) = x² is a surjective function or not.

Solution:

For the given function g(x) = x², the domain is the set of all real numbers, and the range is only the square numbers, which do not include all the set of real numbers. Hence the given function g is not a surjective function.

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Practice Questions on Surjective Function

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FAQs on Surjective Function

What Is Meant by Surjective Function?

A function is a subjective function when its range and co-domain are equal. We can also say that function is a subjective function when every y ε co-domain has at least one pre-image x ε domain.

Can A Function Be Both Injective Function and Surjective Function?

Yes, there can be a function that is both injective function and subjective function, and such a function is called bijective function. Here a bijective function is both a one-to-one function, and onto function. Each value of the output set is connected to the input set, and each output value is connected to only one input value.

What Is the Other Name of Surjective Function?

The other name of the surjective function is onto function. Here every element of the range is connected with at least an element of the domain.

What Is the Relationship Between a Co-domain And A Range In Surjective Function?

The co-domain and a range in a subjective function are the same and equal. Every element of the range has a pre image in the domain set, and hence the range is the same as the co-domain.

How Do You Determine If A Function Is Surjective Using the Graph?

The method to determine whether a function is a surjective function using the graph is to compare the range with the co-domain from the graph. If the range equals the co-domain, then the given function is onto function or the surjective function..

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