**What is a function? **A relation between a group of inputs that each have one output is known as a function. To put it simply, a function is a relationship in which each input has a specific relationship to one output. There is a domain, codomain, and range for each function. The general notation for a function is **f(x)**, where **x** represents the input. A function is generally represented as **y = f(x)**.

## What is a Function in Maths?

A Function in Maths is a unique relationship between inputs (also referred to as the domain) and their outputs (also referred to as the codomain) in which each input has one output that can be linked back to its input.

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## Types of Functions

**One to One Function****Many to One Function****Onto Function****One – One and Onto Function**

### One to One Function

A one to one function is a unique function in which each element of the range is mapped to exactly one element of its domain i.e, the outputs are never repeated.

#### Example

**Let’s use the function f(x)=3x−2**

Assume f(a1)=f(a2)

So, 3a1−2=3a2−2

**Simplify**: 3a1−2=3a2−2 which implies 3a1=3a2

Further simplify: a1=a2

Since a1 must equal a2 when f(a1)=f(a2), this function is indeed one-to-one.

- For x=1 : f(1)=3(1)−2=1
- For x=2 : f(2)=3(2)−2=4
- For x=3 : f(3)=3(3)−2=7

**X-axis:** Represents input values

**Y-axis:** Represents output values f(x).

**Many to One Function**

This function transfers one or more elements from set A to an same element in set B. The same image appears in B for two or more of A’s elements.

#### Example

**f(x)=x ^{2}**

x=2 and x=-2

**For x=2 ,**

f(2)=2^{2}=4

**For x=−2**

f(−2)=(−2)^{2}=4

**Onto Function**

Onto function could be explained by considering two sets, Set A and Set B, which consist of elements. If for every element of B, there is at least one or more than one element matching with A, then the function is said to be **onto function**

#### Example

**f:R→R**

**Domain**: R (all real numbers)

**Codomain:** R (all real numbers)

To show that this function is onto, we need to demonstrate that for every y∈Ry \in \mathbb{R}y∈R, there exists an

x∈R such that f(x)=yf(x) = yf(x)=y

Start with f(x)=y

Substitute f(x)with 2x+1

2x+1=y

**Solve for x:**

2x=y−1

x= y−1 / 2

Since for every y there exists an x (specifically x=y−1 / 2) , **f(x)=2x+1 is an onto function**.

**One – One and Onto Function**

Function, f is One – One and Onto or Bijective if the function f is both One to One and Onto function. In other words, the function f associates each element of A with a distinct element of B and every element of B has a pre-image in A.

**Function:** f:{1,2,3}→{a,b,c} defined by:

f(1)=a

f(2)=b

f(3)=c

- Each element of the domain maps to a unique element in the codomain.

- No two domain elements map to the same codomain element.

**Conclusion:** f is one-to-one.

- Every element in the codomain {a,b,c} is covered.

- a, b, and c are all mapped by 1, 2, and 3, respectively.

**Conclusion:** f is onto.

**Overall:** The function f is both one-to-one and onto.

Read More **What Is A Range In Math?**