Identity

A polynomial function is something like the identity function. It’s a form of a linear function in which the result and input are the same. An identity function, also known as an identity map or an identity relation, 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 definition, its domain, range, graph, and attributes in this course.

What Is an Identity Function?

When a function returns the same value as the output that was used as its input, it is called an identity function. Let’s get started by learning what an identity function is. Every element in the first set has a distinct counterpart in the second. The identity function is a mathematical function that keeps the value of a variable after performing the action specified in the function. An identity transformation is an additional name for an identity function. When used as a parameter, an identity function always produces that very same value.

Identity Function Definition

An identity function returns the image of respectively element in a set B as the equivalent element, e.g., g (b) is equal to b ∀ b ∈ B. As a result, it has the form g(x) = x and is indicated by the letter “I.” Because the image of an element in the domain is matching to the output in the range, it’s termed an identity function. As a result, this function maps an individual real number to itself. An identity function’s output is the same as its input. Because the preimage and the picture are similar, identity functions are easily detected.

Domain, Range, and Inverse of Identity Function

An identity function is a real-valued function that has the form g: R → R and has the value g(x) = x, for each x ∈ 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 √5, the output value is also √5; if the input value is 0, the output value is similarly 0.

• The identity function g(x) has the domain R.

• The g(x) identity function’s range is also R.

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

The domain and range of any function are swapped when it is inverted. This means that the identity function is invertible and has the same inverse as itself.

Identity Function Graph:

After learning about the identity definition, let’s discuss its graph!

The graph of the identity function is always a straight line passing through zero. Every point on the straight line in the graph of an identity function corresponds to the same value on both the ‘x’ and ‘y’ axes. i.e., the function’s values are represented on the Y-axis as y = f (x), while the values of the function’s argument, ‘x,’ are denoted on the X-axis of the coordinate plane. If a point represents a value of ‘4’ on the X-axis, the point likewise indicates a value of ‘4’ on the Y-axis.

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 identical.

Important Notes on Identity Function

Here’s a rundown of some key concepts to keep in mind while studying identity function.

 • The identity functions’ domain and range are the same.

• The identity function graph’s slope is always the same: 1.

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. In the case of identical functions, the input and output are the same.