Types of Functions Based on Set Element

Types of Functions based on Set Elements

Types of functions based on set elements depend on the number of relationships amongst the elements in the domain and the codomain. The different types of functions depending on the set elements are as discussed below.

1. One–One Function or Injective Function

The one-to-one function is also termed an injective function. Here each element of the domain possesses a different image or co-domain element for the assigned function.

A function f: A → B is declared to be a one-one function if different components in A have different images or are associated with different elements in B.

 Example

 Let the function f : R → R be defined by f (x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.


Solution

For any two elements x1, x2 ∈ R such that f (x1) = f (x2), we have
4x1 – 1 = 4x2 – 1
⇒ 4x1 = 4x2, i.e., x1 = x2

2. Onto Function or Surjective Function

A function f: A → B is declared to be an onto function if each component in B has at least one pre-image in A. i.e., If-Range of function f = Co-domain of function f, then f is onto. The onto function is also termed a subjective function.

Example  

Let A = {1, 5, 8, 9) and B {2, 4} and f={(1, 2), (5, 4), (8, 2), (9, 4)}. Then prove f is a onto function.

Solution:

From the question itself we get,

A={1, 5, 8, 9)

B{2, 4}

& f={(1, 2), (5, 4), (8, 2), (9, 4)}

So, all the element on B has a domain element on A or we can say element 1 and 8 & 5 and 9 has same range 2 & 4 respectively.

Therefore, f: A → B is a surjective function.

3. Bijective Function or One One and Onto Function

A function f: A → B is declared to be a bijective function if it is both one-one and onto function. In other words, we can say that every element of set A is related to a different element in set B, and there is not a single element in set B that has been left out to be connected to set A

Example :

Show that the function f : N → N, given by f(x) = x + 1, if x is odd, and x - 1, if x is even, is a surjective function,

Solution:

Let f(x1) = f(x2). Further, let us suppose that x1 is odd, and x2 is even, then we have x1 + 1 = x2 - 1, or x2 - x1 = 2, which is not possible.

Also the possibility of x1 being even, and x2 being odd is also ruled out, using the same argument. Therefore x1, and x2 both should be either odd or even.

Let us assume both x1, and x2 to be odd, and we have f(x1) = f(x2) ⇒ x1 + 1 = x2 + 1 ⇒ x1 = x2. Also, if both x1, and x2 are even, we have f(x1)=f(x2)">

⇒ f(x1) = f(x2) ⇒ x1 - 1 = x2 - 1 ⇒ x1 = x2. Therefore, the function f is a one-one function.

Further, any odd number 2n + 1 in the codomain of N is the image of 2n + 2 in the domain of N, and any even number 2n in the co-domain of N, is the image of 2n - 1 in the domain N. Hence the function is onto function.

Therefore, the given function is a bijective function.

 Example: For A = {−1,2,3} and B = {1,4,9}, f: A→B defined as f(x) = x2 is bijective. 

4. Into Function

Any function f :X → Y is said to be an into function if there exists at least one element in Y which does not have a pre-image in X. i.e., If the Range of function f ⊂ Co-domain of function f, then f is into.

To summarise we can say that the into function is precisely opposite in features to an onto function. That is here certain elements in the co-domain do not own any pre-image. This states that the elements in set Y are excess and are not equated to any elements in set X.

Example:

Sets P = {1, 2, 3} and Q = {5, 8, 9, 10} are defined by the function f = (1, 5), (2, 9), (3, 8).

 Because element 10 of set Q lacks a pre-image in set P, this function is an into function.

 5. Many-One Function

Any function f / A -> B is said to be many- one if two (or more than two) distinct components in A have identical images in B. In a many-to-one function, more than one element owns the same co-domain or image.

                                                   

Example

The set A = {1, 2, 3, 4, 5} as the domain, and the Set B = {x, y, z} as the range.

Here the function f from A to B is said to be many one function, if we have f = {(1, x), (2, x), (3, x), (4, y), (5, z)}.               

6. Constant Function

A constant function is a significant form of a many-to-one function. In this function, all the domain elements possess a single data/ image.

A constant function is the sort of function that presents the same value of output for any presented input. It is represented as, f(x) = c, where c is a constant. For example, f(x) = 2 is a constant function.

OR

The constant function is mathematically expressed as f: R→R and is represented as f(x) = y = c, for x ∈ R and c denotes a constant in R. The domain of the function f signifies R and its range is a constant, c. Sketching a graph, we obtain a straight line that is parallel to the x-axis as displayed above.