Identity Function

An identity function is also known as an identity relation, identity map, or identity transformation. It is one of the many types of functions found in mathematics. It is a function that maps each element in a set to itself, resulting in an output that is identical to the input.

In other words, an identity function ensures that each element in the domain is mapped onto itself in the range, maintaining the equality between the pre-image and image. In this article, we will be learning about all things related to Identity Function.

What Is an Identity Function?

An identity function is a function where each element in a set is mapped to itself, such that the output is identical to the input. In other words, the output of the function is same as input.

For example if input is 2 for identity function then output is also 2. Also, it maintains a one-to-one correspondence between input and output values.

Identity Function Definition

For all elements a ∈ A , if g (a) = a, then g(x) = x is called an identity function.

An identity function has the domain and range of Real numbers. Pre-image and the image of the identity function are identical.

Here f is an identity function as its image and preimage are identical. If the input is 1 then the output is 1, input is 2 then the output is 2, so,

f = {(1,1), (2,2), (3,3), (4,4)}

Examples of Identity Function

Some other function which can be said identity function are:

• f: N → N, f(x) = √(x²)
• g: Z^– → Z^–, g(x) = -√(x²)
• h: N → N, h(x) = |x|
• k: Z^– → Z^–, k(x) = – |x|
• l: N → N, l(x) = [x] (where [o] represent greatest integer function.

Note: All the functions which maps same input to same output are examples of identity function.

Domain and Range of Identity Function

An identity function is a real-valued function that can be represented as f: R → R such that f(x) = x, for each x ∈ R. Here, R is a set of real numbers which is the domain and range of the function f. The domain and the range of identity functions are same.

• Domain: The set of input values or x values of a function f(x).
  • The domain of the identity function f(x) is R

• Range: The set of output values or y values of a function f(x).
  • The range of identity function f(x) is R

Graphical Representation of Identity Function

To plot the graph of an identity function, start by taking the values of x-coordinates on the x-axis and the y-coordinates values on the y-axis. 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.

Consider an example for the identity function, f(x) = x,

Substitute x=-2, -1, 0, 1, 2 in f(x),

x	f(x)
-2	-2
-1	-1
0	0
1	1
2	2

Plot these points and join these points on the graph as shown below,

Slope and Equation of Graph of Identity Function

The straight line makes an angle 45° with x-axis and y-axis

Take the slope intercept form,

y = mx + c

Where m is the slope, c is y-intercept,

Consider two points (x₁, y₁)=(-2,-2), (x₂, y₂ )=(-1,-1),

From the graph the slope(m) of the identity function is,

[Tex]m=\frac{y_2-y_1}{x_2-x_1}[/Tex]

⇒ [Tex]m=\frac{-1-(-2)}{-1-(-2)}[/Tex]

⇒ [Tex]m=\frac{1}{1}[/Tex]

⇒ m=1

As c is y-intercept (the value of y at x = 0)

Put x = 0 and m = 1, in equation

y = x + c

y-intercept is (x, y) = (0,0)

c = y = 0

The equation of the graph in the slope intercept form is,

y = 1x + 0 ⇒ y = x

Properties of Identity Function

Some key properties of the identity function:

• The identity function is a real-valued linear function.
• The domain and range of the identity function are the same.
• The graph of the identity function is a straight line that makes a 45° angle with both the x-axis and y-axis. The slope of the graph is always 1.
• The identity function is a one-to-one and onto function.
• Composing the identity function with itself results in the identity function, i.e. g ∘ g(y) = y.
• The inverse of the identity function is the identity function itself.

Derivative of Identity Function

Derivative of identity function is 1.

As derivative is defined as the rate of change of the function with respect to other variable or independent variable.

Let Identity function f(x) = x,

Use derivative formula,

[Tex]\frac{d}{dx}(x)=1[/Tex]

The derivative of Identity function f(x) with respect to ‘x’ is,

[Tex]\frac{d}{dx}(f(x))=\frac{d}{dx}(x)=1[/Tex]

The derivative of Identity function is ‘1’.

Integral of Identity Function

Integral of identity function is x²/2 + C.

Integral of a function is to find the area under graph of a function for some interval.

Let Identity function f(x)=x,

Use Integral formula,

[Tex]\int x \ dx=\frac{x^2}{2}[/Tex]

The Integral of the Identity function f(x) is,

[Tex]\int f(x) \ dx=\int x \ dx=\frac{x^2}{2} + C[/Tex]

Where C is constant of integration.

How to Identify an Identity Function?

There are many ways to check whether or not any given function is identity or not. Some of these methods are:

• Compare the given function with function definition.
• Plot the function on a graph.
• Substitute some values of x into the function and check if the output matches the input.

Let’s consider an example for better understanding.

Example: Check whether f(x) = 2x is identity function or not.

Solution:

Substitute x = 1 in f(x), we get

f(1)) = 2

As input is not same as output. Thus, f(x) = 2x isn’t a identity function.

Conclusion

Identity Function is a function that returns the same value as the input it was given. It’s also known as an identity map, identity relation, or identity transformation. The various applications of identity functions includes in the linear algebra, matrix operations, used to find the derivatives, integrals, computer programming, software development have wide scope in the future.

Read More,

• Function
• Types of Functions
• Invertible Functions
• Relations and Functions
• One-to-One Functions
• Composition of Functions
• Onto Functions (Surjective Functions)

Solved Examples of Identity Function

Example 1: Check whether f(x)=3x is an identity function or not?

Solution:

For the given function f(x)=3x,

Substitute different x values in f(x),

At x = -2

f(-2) = 3(-2) =-6

At x = -1

f(-1) = 3(-1) = -3

In table format is,

x	f(x) = 3x
-2	-6
-1	-3
0	0
1	3
2	6

As the output is not same as the input, f(x) = 3x is not an identity function.

Example 2: Check whether f(x) = √(x²) is an identity function or not?

Solution:

Substitute different x values in f(x),

At x = 1

f(1) = √(1²) = √1 = 1

At x = -1

f(-1) = √(-1²) = √1 = 1

At x = 2

f(2) = √(2²) = √4 = 2

At x = -2

f(-2) = √(-2²) = √4 = 2

In table format is,

x	√(x2)
-2	2
-1	1
0	0
1	1
2	2

As the output is not same as the input for negative numbers, the given function isn’t an Identity function.

Note: But if we restrict the domain and range of this function to natural numbers. It is an example of identity function.

FAQs on Identity Function

What is an identity function?

An identity function is a mathematical function that returns the same value as its input. In other words, it preserves the identity of the input.

What is the general form of an identity function?

The general form of an identity function is f(x) = x, where x is the input and f(x) is the output.

What is the graph of an identity function?

The graph of an identity function is a straight line passing through the origin (0,0) with a slope of 1. It represents a one-to-one correspondence between the input and output values.

What is the domain and range of an identity function?

The domain and range of an identity function are both the set of all real numbers (R) because every real number is mapped to itself.

Is the identity function reversible?

Yes, the identity function is reversible because every input maps to a unique output and vice versa.

Can identity function be expressed in other forms?

Some other forms of identity function with restricted domain and range are:

• f: N → N, f(x) = √(x²)
• h: N → N, h(x) = |x|