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Identity Function
The identity function is an example of a polynomial function. It is a special type of linear function in which the output is the same as the input. The identity function is also known as an identity map or identity relation.
The domain values are equal to the range values for an identity function. In this lesson, we will learn more about the identity function, its domain, range, graph, and properties with the help of examples.
1.  What Is an Identity Function? 
2.  Domain, Range, and Inverse of Identity Function 
3.  Identity Function Graph 
4.  Properties of Identity Function 
5.  FAQs on Identity Function 
What Is an Identity Function?
A function is considered to be an identity function when it returns the same value as the output that was used as its input. Let's go ahead and learn the definition of an identity function.
Identity Function Definition
An identity function is a function where each element in a set B gives the image of itself as the same element i.e., g (b) = b ∀ b ∈ B. Thus, it is of the form g(x) = x and is denoted by "I". It is called an identity function because the image of an element in the domain is identical to the output in the range. Thus, an identity function maps each real number to itself. The output of an identity function is the same as its input. Identity functions can be identified easily as the preimage and the image are identical.
Consider an example of a function that maps elements of set A = {1, 2, 3, 4, 5} to itself. g: A → A such that, g = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}.
From the abovegiven image, we can see that the function f is an identity function as each element of A is mapped onto itself. The function f is oneone and onto.Domain, Range, and Inverse of Identity Function
An identity function is a realvalued function that can be represented as g: R → R such that g(x) = x, for each x ∈ R. Here, R is a set of real numbers which is the domain of the function g. The domain and the range of identity functions are the same. If the input is √5, the output is also √5; if the input is 0, the output is also 0.
 The domain of the identity function g(x) is R
 The range of identity function g(x) is also R
 The codomain and the range of an identity function are equal sets. ⇒ The identity function is onto.
The inverse of any function swaps the domain and range of that function. This implies that the identity function is invertible and is its own inverse.
Identity Function Graph
To plot the graph of an identity function, we can plot the values of xcoordinates on the xaxis and the values of ycoordinates on the yaxis. The graph of an identity function is a straight line that passes through the origin. For an identity function, the domain and range are the same.
We can see from the abovegiven graph that the straight line makes an angle of 45° with both the xaxis and yaxis. The slope of the identity function graph always remains as 1.Properties of Identity Function
Identity functions are mostly used to return the exact value of the arguments unchanged in a function. An identity function should not be confused with either a null function or an empty function. Here are the important properties of an identity function:
 The identity function is a realvalued linear function.
 The graph of an identity function subtends an angle of 45° with the xaxis and yaxis.
 Since the function is bijective, it is the inverse of itself.
 The graph of an identity function and its inverse are the same.
Check out the following pages related to the identity function
Important Notes on Identity Function
Here is a list of a few points that should be remembered while studying identity function.
 The domain and the range of identity functions are the same.
 The slope of the identity function graph always remains as 1.
Examples on Identity Function

Example 1: If g(y) = (2y+3)/(3y2).Then prove that g ◦ g is an identity function.
Solution:
g(y) = (2y+3)/(3y2)
g ◦ g(y) = g(g(y)) = g((2y+3)/(3y2))
=\(\frac{2\left(\frac{2y + 3}{3y  2}\right)+3}{3\left(\frac{2y+3}{3y 2}\right)2}\)
= (4y + 6 + 9y  6)/(6y + 9  6y + 4)
= 13y/13
= y
Answer: g ◦ g(y) = y. Thus, g ◦ g(y) is an identity function

Example 2: The number of elements in the range of an identity function defined on a set containing nine elements is__
a.) 3^{2}
b.) 3^{4}
c.) 3^{8}
d.) 3^{16}Solution:
The correct option is a.) 3^{2}. If an element is related to itself, then it is called an identity function. That is g(x) = x. So, if the set has 9 elements, then the range of the function will also have 9 = 3^{2} elements.
FAQs on Identity Function
What Is an Identity Function in Algebra?
An identity function is a function where each element in a set B gives the image of itself as the same element i.e., g (b) = b ∀ b ∈ B. Thus, it takes the form g(x) = x and it is denoted by "I". It is called an identity function because the image of an element in the set is identical to the element. The output of an identity function is the same as its input. For example, g(0) = 0 and g(2) = 2. The identity function g(x) = 1x+0 is linear, with the yintercept is 0 and the slope m = 1.
What Is the Value of Identity Function?
A function is considered to be an identity function when it returns the same value as the output that was used as its input. That is, if g is an identity function, then the equality g(x) = x holds for all x.
What Is the Slope of the Identity Function?
The identity function is a linear polynomial function. It is defined as g: R → R such that g(x) = x. The domain is equal to the range for an identity function. The slope of an identity function is m=1 s it makes an angle of 45° with the positive xaxis.
What Is a Constant and Identity Function?
A function is considered to be a constant function if it always returns the same constant value for every input value. For example g(x) =1 is a constant value as its output remains the same irrespective of what the input is. A function is considered to be an identity function when it returns the same output as the input. That is, if g is an identity function, then the equality g(x) = x holds for all x.
What Is the Inverse of an Identity Function?
The inverse of any function swaps the domain and range of that function. For an identity function F(x) = x, its inverse function is F^{1}(x) = x. Thus, it is clear that the identity function is its own inverse.
What Does an Identity Function Look Like?
To graph an identity function, we can plot the values of xcoordinates on the xaxis and the values of ycoordinates on the yaxis. For an identity function, whose range and domain are the same, its graph always appears to be a straight line that passes through the origin. The graph of an identity function is a line that is inclined at an angle of 45° to the positive xaxis and passes through the first and the third quadrant.
What Is the Range of an Identity Function?
The domain and the range of identity function contain the real numbers and they both are the same. For an identity function f(x) = x, if the input is √5, the output is also √5; if the input is 0, the output is also 0.
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