Onto Function

Function is a process that relates elements of one set called domain with the unique elements of another set called the codomain. Functions are thus explained in simple terms as ‘for every set of inputs, there is a unique set of outputs’. For a function to exist, the input and output sets should have some elements which means that they should not be empty sets. Functions are generally represented as f (x) = y. Various types of functions include one to one function, many to one function, onto function, into the function, linear, quadratic and cubic functions, identical functions, algebraic functions, rational functions, even and odd functions, periodic functions and many more. In any function, every element in the domain has one and only one element in the codomain. No element in the domain can relate to two elements in the codomain. However, two elements in the domain may relate to the same element in the codomain. 

What is a Function?

A function in a  mathematical expression is a rule that specifies the relationship between two variables : the independent variable and the dependent variable.

A function is a method or a relationship that connects each member 'a' of a non-empty set A to at least one element 'b' of another non-empty set B.

If each element of B has its preimage in A, the function is onto. A surjective function is another name for an onto function. When calculating the inverse of a function, the concept of onto function is crucial. To determine if a function is onto, we must first understand the information about both sets involved.

If every element of set A has exactly one image in set B, the relation is said to be a function. A function is a relation from a non-empty set B with domain A and no two separate ordered pairs in f having the same initial element.

Types of Functions

1. One-One Function: One element of Domain Set is associated with one element of Co-Domain Set in this function.

2. Many One Functions: The function is said to be many-one functions if there exist two or more than two different domain elements having the same image in the co-domain.

3. Onto Function: Every element of the Co-Domain Set has one pre-image in this function.

4. One Onto Function: Bijective (One-to-One Onto) Function is a function that is both injective (one to one) and surjective (onto).

5. Many-One Onto Function: The function which is both many-one and onto, is called many-one onto function. 

6. Into Function: There is no pre-image in domain X for a function that requires an element from co-domain Y.

7. One-One Into Function: If distinct elements of X have different unique images of Y, the function f is called one-one into function.

8. Many One Into Function: If and only if is both many one and into function, the function f is termed the many-one function.

9. Many One Onto Function: If and only if is both many ones and onto, the function f is termed a many-one onto function.

What is Onto Function?

Any function is said to be onto function, if, in the function, every element of the codomain has one or more relative elements in the domain. Onto function is also popularly known as a surjective function. 

One of the onto function examples is a function which checks whether a given number of inputs is an onto function because for every number in the domain there is a unique element in the output function which is either zero or one. When the codomain element is one, the number in the domain is odd and when the codomain element is zero, the number in the domain is odd. In this example, a no of onto functions elements in the domain will have an output as zero. However, none of the elements in the domain can have both zero and one as an output.  

Given any y value, there is only one x that can be paired with the given y in a one-to-one function. Injective functions are also referred to as such. When operating on the coordinate plane, a function is one-to-one if it passes both the vertical and horizontal line tests (to qualify as a function) (to make it one-to-one).

When the range and codomain of a function are the same, it is said to be onto.

Any function can be broken down into an onto function, surjection, and injection, whereas an onto function, a surjection, and injection can be decomposed from any function.

If there is at least one an A such that f(a) = b, the function f from set A to set B is termed an onto function. Because they are all mapped to some element of A, none of the elements is left out in the onto function.

The number of onto functions from one set to another can be calculated using a formula. We need to make sure that all of the elements of B are utilized in the function from A to B.

Properties of Onto Mapping

Only if the range of a function equals the codomain is it termed an onto function. Some of the most important properties of onto function are listed below.

1. Using the onto function, each element in the codomain will be allocated to at least one value in the domain.

2. There is a right inverse for every onto function. Any function with a right inverse can be classified as an onto function.

3. If the range of f equals the codomain of the function f, the function f is an onto, or surjective, function.

If f: A B is an arbitrary function, then each member of A has an image under f, and all of the images are members of T.

Comparing the range with the codomain is the simplest technique to see if a function is an onto function using the graph. The function is onto if the range equals the codomain. If and only if every horizontal line contacts the graph at least one or more points, a graph of any function can be considered onto. If a function's range contains an element that fails the horizontal line test by not crossing the function's graph, the function is not surjective.

Relationship between Function

The one-to-one function is also a requirement for understanding inverse functions. Onto and one-to-one functions are referred to as Surjective and Injective functions, respectively. The main difference is that onto functions affect all of the output values, whereas one-to-one functions affect only one of the output values.

The bijective function is a function that is both One to One and Onto. The output set's values are all related to the input set, and each output value is only associated with one input value.

Numbers of Onto Functions

The numbers of onto functions possible between two non-empty sets is given by the formula:

In the above equation, n is the number of elements in set A and m  is the number of elements in Set B. The first, second, third ……. elements of set B or codomain are represented as m0, m1, m2,.. When the number of elements in set A is less than the number of elements in set B, i.e., if n<m, the number of possible onto functions is zero. If the number of elements in set A and set B are equal i.e., n = m, the number of possible onto mappings is given by m!. No of onto functions for a given domain and range is given as

Where ‘m’ is the range and ‘n’ is the domain

Onto Function Examples

1. Consider a function in which 10 students of a class represent the domain and their roll numbers represent the codomain. State whether these two sets explain ‘what is onto function’ or not. Justify your answer.

Solution: In the above example, Set  A = 10 students of a class and Set B = Roll numbers of the 10 students. Here every element in set  A will have a unique relative element in Set B. This function does not come under ‘onto function’ examples because no two students can have the same roll numbers. i.e., no two elements can have the same relative codomain element. This contradicts the onto function definition. The example stated above is depicted in the figure below.

(Image will be Uploaded soon)

Fun Facts About Onto Mapping

• Every onto function definition is an inverse function and every inverse function is an onto function.

• In any kind of function, several inputs may have the same output whereas one input cannot have two outputs. The output function for each onto the mapping function is unique. 

• If a function is both injective and surjective, then it is called a bijective function.