Learn US

Identity Function, Constant Function

1.2k views

25th September 2020

Reading Time: 6 Minutes

Introduction

Functions are a special type of relation that operates from a non-empty set A to another non-empty set B such that no two distinct ordered pairs in the function have the same first element. It is denoted by the symbol ‘f’. The concept of functions is very important since it showcases the idea of mathematically precise correspondence from one quantity to the other.


Identity Function

Like we have, identity in mathematical operations (additive identity and multiplicative identity), we have identity in functions too! Let us understand about these unique functions better.

What is an identity function?

An identity function is a function wherein each element of set A has an image on itself. Thus, it takes the form f(x) = x. The identity function thus maps each real number to itself. The output here is essentially the same as the input. The domain and range of the function are the same here. If the domain is √5, the range is also √5; if the domain is 0, the range is also 0. It is quite simple to identify an identity function.

Example

Let the elements of Set A be 1, 2, 3, 4. For the function to be an identity function, the elements of Set B has to be the same.

Figure 3


Domain, Range, and Inverse of Identity Function

Identity function is a real-valued function that can be represented as f: R R, y = f(x) = x, where x R.

Here, the domain of f is R
          the range of f is also R, 
          co- domain and range are equal set.

Since the inverse of any function swaps the domain and range of the function, it is clear that the identity function is its own inverse.


Identity Function Graph

As we plot the domain and range of an identity function on the x-axis and y-axis respectively, we observe that the identity function graph is a straight line passing through the origin.

Figure 4

The straight line makes an angle of 45° both with the x-axis and the y-axis. The slope of the identity function graph is always 1.


Properties of Identity Function

The important properties of an identity function can be listed as follows:

  • It is a real-valued linear function.
  • Its graph subtends an angle of 45° with the x and y-axes. 
  • Since the function is injective, it is the inverse of itself.
  • The graph of an identity function and its inverse are the same.

Constant Function

What is a constant in math?

Constant functions take their name from a mathematical constant which is a figure whose value is constant. Let us understand this concept in detail.

What is a constant function?

Have you ever visited a fixed value shop, where everything in the shop has a fixed value? If yes, you have discovered a relation where the things in the shop and their price are an example of a constant function. Defining in terms of mathematics, we can say that a constant function is a function that has the same range for all values of its domain. Thus, it takes the form y = c,

 where c is a constant.

For example,  y = 10 is a form of constant function.

Example

In math, we can find an example of constant functions in exponents. Let us see how.

Let the elements of Set A be  20, 30, 40. For each element of set A, the value in Set B will always be 1.

Example 1


Domain, Range, And Inverse of Constant Function

A constant function is a real-valued function and can be represented as
                                      f: R   R, y = f(x) = c, x  R. 
Here, the domain of f is R and its range is c, where c can be any real number.

 Let f(x) = 2, be a constant function. Thus, its domain will be the set of Real numbers and range 2.

 Let us now find its inverse. For this, we will replace f(x) with y, so that we now have y = 2. Swapping the roles of x and y we now have x = 2, which is not a function since it defies the fundamental definition of a function (A relation

 is defined as a function if every element of set A has one and only one image in set B).  Hence,  constant functions don't have an inverse. 


Constant Function Graph

On a Cartesian plane, the constant function graph is always a straight line parallel to the x-axis. It will be above the x-axis, in case the value of range is positive; below the x-axis, in case the value of range is negative; and be coincident with the x-axis in case the range is 0.  It goes without saying that the slope of any constant function graph is always 0.

Figure 1

Figure 2

Figure 3


Properties of Constant Function

Basic properties of constant function can be listed as follows:

  • It is a linear, real-valued function.
  • The graph of a constant function is always a horizontal line parallel to the x-axis that passes through the coordinates(0,c).
  • It is an even function as its graph is symmetric with respect to the y-axis.

What is the slope of a constant function?

The slope of the line of a constant function graph is 0. This can be verified by observing how it is parallel to the x-axis. 


Summary

  • Both constant and identity functions are real-valued and linear in nature.
  • While a constant function is a function that gives the same output for every value of input, an identity function gives the same output as its input.
  • Whereas the graph of a constant function is a straight line parallel to the x-axis, with its slope = 0; the graph of identity function is a straight line passing through its origin, with its slope=0.
  • Constant function takes the form f(x) = c; identity function has the form f(x) = x.
  • A constant function example is y = 7 while an identity function example is y = x. 

FAQs

What is the identity of a function?

Answer: An identity function is a type of function that returns the same value as its argument. Hence, the domain and range take the same value in an identity function.

What is the constant of a function?

Answer: A constant function is a type of function that returns the same value for every argument. Hence, for every domain, the range is always the same or constant.

What are the three types of functions?

Answer: The three type of functions are- 

  • Linear functions
  • Quadratic functions
  • Exponential functions 

What are the examples of functions in real life?

Answer: We see many examples of functions in real life like-

  • Shadow
  • Reflection
  • Vending machine
  • Geometric patterns
     

GIVE YOUR CHILD THE CUEMATH EDGE
Access Personalised Math learning through interactive worksheets, gamified concepts and grade-wise courses
Learn More About Cuemath