The identity function is an example of a polynomial function. It is a special type of linear function in which the output is the same as the input. The identity function is also known as an identity map or identity relation.
The domain values are equal to the range values for an identity function. In this lesson, we will learn more about the identity function, its domain, range, graph, and properties with the help of examples.
| 1. | What Is an Identity Function? |
| 2. | Domain, Range, and Inverse of Identity Function |
| 3. | Identity Function Graph |
| 4. | Properties of Identity Function |
| 5. | FAQs on Identity Function |
A function is considered to be an identity function when it returns the same value as the output that was used as its input. Let's go ahead and learn the definition of an identity function.
An identity function is a function where each element in a set B gives the image of itself as the same element i.e., g (b) = b ∀ b ∈ B. Thus, it is of the form g(x) = x and is denoted by "I". It is called an identity function because the image of an element in the domain is identical to the output in the range. Thus, an identity function maps each real number to itself. The output of an identity function is the same as its input. Identity functions can be identified easily as the pre-image and the image are identical.
Consider an example of a function that maps elements of set A = to itself. g: A → A such that, g = <(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)>.
From the above-given image, we can see that the function f is an identity function as each element of A is mapped onto itself. The function f is one-one and onto.
An identity function is a real-valued function that can be represented as g: R → R such that g(x) = x, for each x ∈ R. Here, R is a set of real numbers which is the domain of the function g. The domain and the range of identity functions are the same. If the input is √5, the output is also √5; if the input is 0, the output is also 0.
The inverse of any function swaps the domain and range of that function. This implies that the identity function is invertible and is its own inverse.
To plot the graph of an identity function, we can plot the values of x-coordinates on the x-axis and the values of y-coordinates 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.
We can see from the above-given graph that the straight line makes an angle of 45° with both the x-axis and y-axis. The slope of the identity function graph always remains as 1.
Identity functions are mostly used to return the exact value of the arguments unchanged in a function. An identity function should not be confused with either a null function or an empty function. Here are the important properties of an identity function:
Check out the following pages related to the identity function
Important Notes on Identity Function
Here is a list of a few points that should be remembered while studying identity function.
Example 1: If g(y) = (2y+3)/(3y-2).Then prove that g ◦ g is an identity function. Solution: g(y) = (2y+3)/(3y-2) g ◦ g(y) = g(g(y)) = g((2y+3)/(3y-2)) =\(\frac<2\left(\frac<2y + 3>\right)+3><3\left(\frac<2y+3>\right)-2>\) = (4y + 6 + 9y - 6)/(6y + 9 - 6y + 4) = 13y/13 = y Answer: g ◦ g(y) = y. Thus, g ◦ g(y) is an identity function
Example 2: The number of elements in the range of an identity function defined on a set containing nine elements is__
a.) 3 2
b.) 3 4
c.) 3 8
d.) 3 16 Solution: The correct option is a.) 3 2 . If an element is related to itself, then it is called an identity function. That is g(x) = x. So, if the set has 9 elements, then the range of the function will also have 9 = 3 2 elements.
Math will no longer be a tough subject, especially when you understand the concepts through visualizations.
