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: 

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 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:

Identity Function Graph

The identity function graph is always a straight line that 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 are indicated on the Y-axis as y = f (x) and the values of the argument used in the function i.e. ‘x’ are 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.

(Image will be uploaded soon)

Identity Function Properties

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

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

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

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

• The 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. 

Solution:

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.

(Image will be uploaded Soon)

Identity Function Fun Facts:

• The 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 a slope equal to 1 and y-intercept equal to (0, 0)

Domain, Range, and Inverse of Identity Function

An identity function is a real-valued function of the form g: R→R such that g(x) = x for any x ∈ R. R denotes the domain of the function g, which is a set of real numbers. Identity functions have the same domain and range. If the input is 5, the output will be 5 as well; if the input is 0, the output will be 0. 

• R is the domain of the identity function g(x).

• R is also the range of the identity function g(x).

• An identity function's co-domain and range are equal sets, and the identity function is onto.

The inverse of any function flips the function's domain and range. This means that the identity function may be inverted and is its own inverse.

Constant Function

A constant function is used to express a number that remains constant across time, and it is regarded as the simplest sort of real-valued function. Constant functions are linear functions with graphs in the plane that are horizontal lines. One of the real-life examples of constant functions is the maximum number of points that may be achieved in an examination. A constant function is one that has the same range for different domain values. A constant function is represented graphically as a straight line parallel to the x-axis. The function's domain is the x-value, which is displayed on the x-axis, and the function's range is y or f(x), which is denoted with reference to the y-axis. Any function is a constant function if it has the form y = k, where k is a constant and k can be any real number. f(x) = k is another way to write it.

Constant Function in Real Word

There are several instances in real life where constant functions are used. Constant functions are employed in this case to depict scenarios in which one parameter is constant and independent of the other independent values. Here are some real-world instances of constant functions:

Every item in a department shop costs $3.

An exam in which every student received a star regardless of how hard they all worked.

For all orders above $300, a $30 bag is included.

A school canteen where every child, regardless of grade or age, received one sandwich.