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.
ONE to ONE FUNCTION

Many – One Function

Many to 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,
Illustration 2 for Onto________function_Surjective_Function

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}
Illustration 3 for Onto________function_Surjective_Function

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.
Linear Functions

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.