Types of Functions

There are three types of functions;

  1. One-one or Injective function
  2. Onto or Surjective function
  3. One-one onto or Bijective function

One-one or Injective Function

A function ƒ: A➞B is said to be one-one or injective, if distinct elements (or pre-images) in A have distinct images in B.

For any x, y ∈ A, \[x ≠ y \Rightarrow ƒ(x) ≠ ƒ(y); \] or, equivalently, \[ƒ(x)=ƒ(y) \Rightarrow x=y\] In other words, a function ƒ is said to be one-one or injective if \[(x, ƒ(x)), (y, ƒ(y)) ∈ ƒ \Rightarrow x=y\] Thus, under one-one function all elements of A are related to different elements of B.

Examples:

  1. The function ƒ: Z➞Z (Z is a set of integers) defined by ƒ(x)=2x is one-one, since \[ƒ(x)=ƒ(y) \Rightarrow 2x=2y \Rightarrow x=y\] It can be diagrammatically represented by
One-one or Injective Function: One-one and into

Here, \[ƒ(Z)=\{…, -4, -2, 0, 2, 4, …\}⊂Z.\] In such a case, the function is said to be one-one and into.

A function ƒ: A➞B is said to be into function if at least one element of B has no pre-images in A.

2. The function ƒ: R➞R (R is the set of real numbers) defined by ƒ(x)=x3 is one-one, since the cubes of different real numbers are themselves different.

3. The function ƒ: N➞R (N is the set of natural numbers) defined by ƒ(x)=x2, is one-one and into, since the squares of different natural numbers are themselves different and ƒ(N)⊂R.

4. The function ƒ: R➞R defined by ƒ(x)=x2, is not one-one, since ƒ(2)=ƒ(-2)=4, i.e. the images of two numbers -2 and 2 have a same image 4.

Onto or Surjective Function

A function ƒ: A➞B is said to be onto or surjective function, if every element of B is an image of at least one element of A i.e. every element of B has a pre-image. Then, \[ƒ(A)=B\]

Examples:

  1. Let A={-3, -2, -1, 1, 2, 3} and B={1, 4, 9}. Then, ƒ: A➞B defined by ƒ(x)=x2 is onto.
Onto or Surjective Function: Arrow diagram, many-one onto fuction

Here, ƒ(A)=B. This function is many-one onto.

A function ƒ: A➞B is said to be many-one if two or more elements of set A have a same image in set B.

2. A function ƒ: A➞B such that ƒ(x)=c ∈ B for every x ∈ A is a many-one onto function. Such a function is called a constant function. \[ƒ:A=\{-3,-2,-1,1,2,3\}\rightarrow B=\{0\}\]

Onto or Surjective Function: Arrow diagram, constant function

3. Let A={a, b, c} and B={1, 2}, then,

Onto and Into Fuction

4. The function ƒ: Z➞N, where Z is the set of integers and N is the set of natural numbers, defined by ƒ(x)=x2 is not onto, since there is no x∈Z s.t. ƒ(x)=x2=2. There are several pairs of values of x which has the same image. ƒ(Z)≠N but ƒ(Z)⊂N. Hence, this is a many-one into function.

One-one Onto or Bijective Function

A function that is both one-one and onto (i.e. injective and surjective) is called a bijective function. It is also known as a one-to-one correspondence.

A bijective function from a set A to itself ƒ: A➞A is known as a permutation or operator on A.

Examples:

  1. Let A={Earth, Jupiter, Saturn} and B={Moon, Io, Titan}. Then, the following diagram shows a bijective function.
One-one onto or Bijective Function

2. The function ƒ: A➞B defined by ƒ(x)=x for all x∈A is a bijective function. This function is also known as an identity function.

3. The function ƒ: N➞{3} defined by ƒ(x)=3 for all x∈N is not one-one but onto. So, it is not a bijective function.

More on Relations And Functions

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