# Types of Functions: One – One Function: Many – One Function (For CBSE, ICSE, IAS, NET, NRA 2022)

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# Title: Types of Functions

## Types of Functions

- One – one function (Injective function)
- Many – one function
- Onto – function (Surjective Function)
- Into – function
- Polynomial function
- Linear Function
- Identical Function
- Quadratic Function
- Rational Function
- Algebraic Functions
- Cubic Function
- Modulus Function
- Signum Function
- Greatest Integer Function
- Fractional Part Function
- Even and Odd Function
- Periodic Function
- Composite Function
- Constant Function
- Identity Function

## One – One Function (Injective Function)

- If each element in the domain of a function has a distinct image in the co-domain, the function is said to be one – one function.
- A function in which one element of Domain Set is connected to one element of Co-Domain Set.

## Many – One Function

On the other hand, if there are at least two elements in the domain whose images are same, the function is known as many to one function

## Onto – Function (Surjective Function)

- A function is called an onto function if each element in the co-domain has at least one pre – image in the domain.
- A function in which every element of Co-Domain Set has one pre-image.
**Example**: Consider,

## Into – Function

- If there exists at least one element in the co-domain which is not an image of any element in the domain then the function will be Into function.
- A function in which there must be an element of co-domain Y does not have a pre-image in domain X.
- For different values of Input, we have different output hence it is one – one function also it manage is equal to its co-domain hence it is onto also.
- Consider,

- In the function f, the range i.e.. , {1,2, 3} ≠ co-domain of Y i.e.. , {1,2, 3,4}

## Polynomial Function

- A polynomial function is defined by where n is a non-negative integer and .
- The highest power in the expression is the degree of the polynomial function. Polynomial functions are further classified based on their degrees:
- The highest power in the expression is known as the degree of the polynomial function. The different types of polynomial functions based on the degree are:
- The polynomial function is called as Constant function if the degree is zero.
- The polynomial function is called as Linear if the degree is one.
- The polynomial function is Quadratic if the degree is two.
- The polynomial function is Cubic if the degree is three.

## Linear Function

- All functions in the form of where are called as linear functions. The graph will be a straight line.
- In other words, a linear polynomial function is a first-degree polynomial where the input needs to be multiplied by m and added to c.
- It can be expressed by
- Taking into consideration, . The domain and the range are
**R**. The graph is always a straight line.

## Identical Function

- Let
**R**be the set of real numbers. If the function is defined as , then the function is known as Identity function. The domain and the range being**R**. - The graph is always a straight line and passes through the origin.

## Quadratic Function

- All function in the form of where, will be known as Quadratic function. The graph will parabolic.
- In simpler terms,
- A Quadratic polynomial function is a second degree polynomial and it can be expressed as;
- , and a is not equal to zero.
- Where a, b, c are constant and x is a variable.