Identity Function

Identity function belongs to the category of polynomial functions. It is a function whose input and output are the same. It is also known as identity relation or identity map. The identity function for any set A is given as:

g (a) = a ∀ a ∈ A

This implies, g (x) = x

This means for every element in A, the value of ‘a’ will be ‘a’ only. Thus, an identity function is the real function of itself.

Consider an example of a set of numbers. We need to map the function of set A = {1, 2, 3, 4, 5} to itself. Then,

g: A → A such that, g = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}.

From the identity function, each element will map its corresponding element only. Therefore, the identity function will be represented as:

Domain, range and inverse

Domain – Since identity function is a real-valued function, it can be represented as: g: R → R such that g(x) = x, for every x ∈ R. where, R = set of real numbers.
Range – The domain and range of identity functions are the same. For example, we gave 5 rupees to the shopkeeper; then, he will return 5 rupees only if the function is an identity function.
Inverse – The inverse of a function means swapping the values of its domain and range. Since the domain and range of the identity function are the same, the inverse is the same as the original function.

Identity function graph

We can plot the graph of the identity function if we know the values of x and y coordinates. For example, by knowing x and y coordinates, both positive and negative, we can mark those points and join a line to make the graph as shown below:

Properties of the identity function

Identity functions are mostly used to find the exact values of an unchanged function. It should not be confused with a null or empty function. All the three are different. Some of the properties of identity function are:
1. The identity function is real-valued.
2. The graph of identity function makes an angle of 45 degrees with the x and y axes.
3. Since the identity function is bijective in nature, the inverse of the identity function will be the same as the original.
4. Also, the graph of the inverse function will be the same as the original one.

Example: Prove f (2x) = 2x is an identity function.

Solution:

To prove, let us put the values of x in the given function.
If x = 1, then;
f (2(1)) = 2(1)
Therefore, f (2) = 2
If x = 2,
then; f (2(2)) = 2(2)
Therefore, f (4) = 4
If x = 3, then;
f (2(3)) = 2(3)
Therefore, f (6) = 6
If x = 0, then;
f (2(0)) = 2(0)
Therefore, f (0) = 0
Let us find some negative values of x.
If x = -1,
then; f (2(-1)) = 2(-1)
Therefore, f (-2) = -2
If x = -2,
then; f (2(-2)) = 2(-2)
Therefore, f (-4) = -4
If x = -3,
then; f (2(-3)) = 2(-3)
Therefore, f (-6) = -6

We can draw the table as: