Onto Function: Overview, Questions, Preparation

A mathematical function is an expression that describes the relationship between the elements of one set with the elements in another set. Functions are crucial as they aid in explaining and deducing various physical relations in scientific expression.

Onto function is a subtype of function just like the one-one function and many one functions. Onto functions are also known as surjective functions, and the term was coined by a French alumnus of the École normale supérieure, Nicolas Bourbaki. 

Definition of Onto Function 

The onto function is a type of function, where elements of one set find at least one or more elements in another set. Moreover, a surjective function is a function whose representation is equal to its co-domain. Function f with domain X and co-domain Y is similarly surjective if there is at least one x in X with f(x)=y for every y in Y.

Any function induces a surge by limiting its co-domain to the representation of its domain. Every function with a right inverse is primarily a surjection where every surjective function has a right inverse. The formulation is always surjective for an onto function. Such functions can be disintegrated into injection and surjection.

Properties of Onto Function

Following are the properties of onto functions:

1. There is a right inverse for every onto function.
2. The decomposition of an onto function always yields an onto function.
3. The elements in a co-domain is restricted to the function range in an onto function.
4. The domain is essentially what will move into the function, the co-domain notes possible outcomes and the sector reflects the successful results of the function.

Formula for Calculating the Number of Onto Functions

The formula for calculating the number of onto function is mⁿ – m + m ((m - 1)ⁿ – (m – 1)).
This formula is used to calculate the onto function for n number of elements in one set to another set containing m number of elements. The number of onto function is zero when n

The topic of onto functions is from the chapter relations and functions. The chapter holds various other important topics such as types of relations, one-one function, and many one functions. The chapter accounts for valuable 8 marks in the 12^th final examination.

1. f: R → R defined by f(x) = 3 − 4x. Find whether the function is bijective, surjective, or injective.
Solution.
Given,  f: R → R defined by f(x) = 3 − 4x, where x and y belong to R. 
Therefore, F(x) = F(y)
3 - 4x = 3 - 4y
-4x = -4y
x = y
Therefore, F(x) is injective.
Also, there is (3-y)/4 in R for every real number (y)
f(3-y/4) = 3 - [4(3-y/4)]
= y
Therefore, F(x) is bijective.

Q: How is the feature of onto function graphs?

Q: What is the other name of an onto function?

Q: What constitutes the term ‘bijective’?

Q: What are the four different types of function?

Q: How do you know if the surjections are epimorphisms?