Identity function

An identity function, also known as an identity relation, identity map, or identity transformation in mathematics, is a function that always returns the same value as its argument.

A polynomial function is something like the identity function. It’s a form of linear function in which the result and input are the same. The identity function which is also known as the identity map or the identity relation that is a type of identity function.

For an identity function, the domain values are the same as the range values. With the use of examples, we’ll learn more about the identity function, its domain, range, graph, and attributes in this course.

Identity function:-

When a function returns the same value as the output that was used as its input, it is called an identity function. 

Definition:-

An identity function is a function that returns the image of each element in a set B as the same element, identity function  is indicated by the letter “I.” Because the image of an element in the domain is identical to the output in the range, it’s termed an identity function. As a result, an identity function is a function that maps each real number to itself. An identity function’s output is the same as its input. Identity functions may be identified easily as the pre-image and the image are identical.

In other words, the codomain L’s function result f(X) is always the same as the domain L’s input element X. The identity function on L is bijective since it is both an injective and a surjective function. 

Range ,domain and inverse of identity function:-

An identity function is a real-valued function that has the form g: R  to R and has the value x for each x belong to R. The domain of the function g is represented by R, which is a collection of real numbers. Identity functions have the same domain and range as each other. If the input value is  9, the output value is also 9; if the input value is 2, the output value is similarly 2.

• The identity function g(x) has the domain R.
• R is also the range of the identity function g(x).
• An identity function’s co-domain and range are both equal sets. Onto is the identity function.

The domain and range of any function are swapped when it is inversed. As a result, the identity functions are  invertible and it has its  own inverse.

Graph of identity function:-

We can plot the values of x-coordinates on the x-axis and the values of y-coordinates on the y-axis to plot the graph of an identity function. An identity function’s graph is a straight line passing through the origin. The domain and the range of the identity functions are the same.

Example of the graph of identity function:-

If we plot the value of x-coordinates on the x-axis and y co-ordinates on the y-axis. As we know the graph of the identity function is a straight line. So, let us let an example of  identity  function f(x)=x

Now, if we put the value of x in the function we get the value of y, and the graph of this function is a straight line.

Properties of identity function:-

Identity functions are commonly used to return the precise value of a function’s arguments, unmodified. A null function or an empty function should not be mistaken for an identity function. An identity function has the following crucial properties:

• The identity function is a linear function with real values.
• The x-axis and y-axis intersect at a 45° angle in the graph of an identity function.
• The function is the inverse of itself because it is bijective.
• An identity function’s graph and its inverse are identities.

Fun facts about identity function:-

• A number’s additive identity is ‘0.’ When a number is multiplied by zero, the sum remains the original value. 
• The number itself is the product of any number multiplied by ‘1’, hence ‘1’ is the multiplicative identity of a number. 
• The slope of an identity function is always 1 and the y-intercept is equal to (0, 0).

Conclusion

The identity function is a function that returns the same value as the argument it was given. It’s also known as an identity map, identity relation, or identity transformation. Graphs can also be used to determine the domain and range of functions. The domain of a graph is made up of all the input values represented on the x-axis since the domain refers to the set of possible input values. The range represents the collection of possible output values on the y-axis.