Functions and Its Types

Types Of Functions

There are 8 types of Functions In Engineering Mathematics. These are as follows:

• Injective (One-to-One) Functions: 
  When one element of Domain Set is associated with one element of Co-Domain Set. It is called an Injective Function.

• Surjective (Onto) Functions: 

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

For Example: Consider, A = {A,B,C,D}, B = {12,24,50} and f = {(A,24), (B,12), (C,50), (D,50)}.

It is a Surjective Function, as every element of B is the image of some A.

• Bijective (One-to-One Onto) Functions: 

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

Here, from the figure below, it is pretty clear that f is a one-to-one function, and also it is onto. So it is a bijective function.

• Into Functions: 
  A function that requires a co-domain Y element and does not have a pre-image in domain X.
  For example: Suppose, A = {A,B,C}  B = {12,24,50,100}   and f: A → B such that  f = {(A, 12), (B, 24), (C,50)}  
  In the function f, the range i.e., {12,24,50} ≠ co-domain of Y i.e., {12,24,50,100}  
  So, it is an into function
   

• One-One Into Functions: 
  When different elements of X have different unique images of Y, the function is called one-one into function.
  For Example: Considering, A = {A,B,C}  B = {12, 24, 50, 100} and f: A → B such that f = {(A, 12), (B, 50), (C, 100)}  
  The function f is a one-one into function

• Many-One Functions: 
  If there are two or more separate elements in X that have the same image in Y, the function f is said to be a many-one function.
  For Example: onsider A = {10, 20, 30, 40, 50}  B = {A, B, C} and f: A → B such that  f = {(10, A), (20, A), (30, A), (40, B), (50, C)}  
  The function f is a many-one function

• Many-One Into Functions: 
  For a function f: X → Y. if and only if f is both many one and into the function it is called the many-one Into function.
  For Example: A = {10, 20, 30} B = {A , B} and f: A →B such that  f = {(10, A), (20, A), (30, A)}

• Many-One Onto Functions: 
  For a function f: X → Y. if and only if ‘f’ is is both many-one and onto the function is called the Many-One Onto Function.
  In the example figure given below, The function f is a many-one (as the two elements have the same image in Y) and is onto (as every element of Y is the image of some element X). So, it is many-one onto function.

FAQs:

1. What do you mean by Invertible Functions?
   Invertible functions are also known as Inverse Functions. A function f: A → B is inverse if and only if it is a bijective function.
2. How to denote functions as a set?
   Considering X and Y as two non-empty sets, then a function ‘f’ from X to Y is a subset of X x Y, with two important constraints:

• ∀ a ∈ X, (a, b) ∈ f for some b ∈ Y
• If (a, b) ∈ f and (a, c) ∈ f then b = c.

Key Takeaways

In this article, we have extensively discussed the ‘Functions and it’s types in Engineering Mathematics. Check out the next topic Identity Functions. 

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