Identity Function

What is an Identity Function?

A function in Mathematics is a binary relation between two sets. Every element in the first set has a unique representative in the other set. Any mathematical function that retains the value of the variable even after the operation indicated in the function is performed is called the identity function. An identity function is also called an identity relation or identity transformation or identity map. An identity function always returns the same value that is used as an argument. An identity function is represented in general as: 

f (x) = x

Identity Function Definition:

For a set of real numbers in Mathematics represented as “R’, the real-valued function f: R → R is represented as y = f (x) = x for all x ∈ R, then y = f (x) is called as an identity function because it returns the value same as that of the argument used. In this function, the values of domain, codomain, and range all are the subsets of the set ‘R”. The graph obtained by plotting an identity function on a coordinate plane is a straight line passing through the origin. Identity functions are used in representing identity matrices. An identity function can be mathematically represented as:


f (c) = c ∀ c ∈ R


Identity Function Graph:

The identity function graph is always a straight line which passes through zero. Every point on the straight line in the graph of an identity function corresponds to the same value on both ‘x’ and ‘y’ axes. i..e. The values of the function is indicated on the Y-axis as y = f (x) and the values of the argument used in the function i.e. ‘x’ is denoted on the X-axis of the coordinate plane. If the value indicated by a point is ‘4’ on the X-axis, the point also indicates the value equal to ‘4’ on the Y-axis. The figure below represents the examples of an identity function graph which denotes the function y = f (x) = x.

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Identity Function Properties:

  • Identity function examples always return the value of its argument. 

  • In the case of vector spaces, the identity function is an application of linear operators.

  • Identity function is a multiplicative function of all positive integers. It indicates the multiplication of a number with 1. 

  • Identity function is represented in the form of an identity matrix in case of m – dimensional vector spaces.

  • Identity function is a continuous function in topological space. 

Identity Function Examples:

  1. Show that the function f (3r) = 3 r is an identity function. Also draw an identity function graph. 


The given function is f (3r) = 3 r. Recall what an identity function is. 

For r = 0,

f (3r) = 3 r

f (3 x 0) = 3 x 0

f (0) = 0 

For r = 1, 

f (3r) = 3 r

f (3 x 1) = 3 x 1

f (3) = 3

For r = 2,

f (3r) = 3 r

f (3 x 2) = 3 x 2

f (6) = 6  

Similarly, it can be shown that for all values of ‘r’ f (3r) = 3r. So, the given function is an identity function. The graph can be plotted using the following table.








y = f (3x)







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Identity Function Fun Facts:

  • Additive identity of a number is ‘0’. This means that the sum remains the number itself when zero is added to any number.

  • The product of any number multiplied with ‘1’ is the number itself and hence ‘1’ is the multiplicative identity of a number. 

  • An identity function always has slope equal to 1 and y-intercept equal to (0, 0)


FAQs (Frequently Asked Questions)

1. What is a Mathematical Function? 

A function in Mathematics is a relation between two non-empty sets called input and output sets. For a specified function, every element in the input set has a representative in the output set. Each input has a specific output. A single element in the output set of a function can represent two or more elements in the input set. However, no element in the input set can have two representatives in the output set. This is one of the basic conditions of a Mathematical function. There are different types of mathematical functions classified based on various factors. Linear function, quadratic function, logarithmic function, identity function are a few examples of Mathematical functions. 

2. What is Identity Function?

Identity function is a function that returns the same value as that of the argument used in the function. i.e. in an identity function, the value of the operand itself is the output of the function. So, input and output are the same in the case of identity functions. An identity function is generally represented as f (a) = a. In this function, the value of the output and the argument used in the function are the same i.e. ‘a’. The graph of an identity function is generally a straight line passing through the origin. Identity function examples are multiplicative functions for positive integers.They indicate the multiplication of a number with 1. 

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