# 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
• 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.