Domain Range and Codomain Of A Function

We are familiar with Domain of a Function and Range of a Function. But what does it mean? before diving deeper in to the topic, let us understand what is a function?

Functions:

Functions are one of the very important concepts in mathematics which has got numerous applications in real world. Be it the mega skyscrapers or super-fast cars, their modeling requires methodical application of functions. Almost all the real world problems are formulated interpreted and solved using functions.

An understanding of relations is required in order to understand functions. And understanding of Cartesian products is required to understand relations. A Cartesian product of two sets \(A\) and \(B\) is collection of all the ordered pairs \((a,b)\) such that \(a ∈ A\) and \(b ∈ B\) . A relation is a subset of a Cartesian product. Hence, a relation is a rule that “relates” an element from one set to an element from another set. A function is a special kind of relation. Let us consider a relation \(F\) from set \(A\) to \(B\).

Definition 1: A relation \(F\) is said to be a function if each element in set \(A\) is associated with exactly one element in set \(B\).

To understand the difference between relations and functions, let us take an example. Set \(A\) contains the name of all the countries that have won cricket world cup and set \(B\) contains the list of years in which world cup was played. The arrow diagram in the fig.1 represents a relation \(R\) but not a function. This is because elements in set \(A\) are associated to more than one element in set \(B\).

Domain And Range Of A Function

But if we define a relation \(F\) from set \(A\) to \(B\) such that it associates the countries with the year they won world cup for the first time, for every element in set \(A\), we will have exactly one association in set \(B\). This relation \(F\) shown in fig. 2 qualifies to be a function.

Domain And Range Of A Function
Domain and Range of a Function:

Remember that in case of a relation, the domain might not be the same as the left set in the arrow diagram. This is because the set may contain any element which doesn’t have an image in the right set. But in case of functions, the domain will always be equal to the first set. Range and Codomain of a function are defined in the same way as they are defined for relations.

Domain and Range

Let’s learn about Domain and Range in detail here.

Domain

  • The set of all possible values which qualify as inputs to a function is known as the domain of the function or It can also be defined as the entire set of values possible for independent variables.
    • Domain can be found if – the denominator of the fraction is not equal to zero and the digit under the square root bracket is positive. (In case of a function with fraction values).

For e.g. the domain of the function F is set A i.e. {India, Pakistan, Australia, Sri Lanka}.

How to find the Domain of a function

  • To find the domain we need to look at the values of the independent variables which are allowed to use as explained above i.e no zero at the bottom of fraction and no negative sign in square bracket.

Range

  •  The set of all the outputs of a function is known as the range of the function x or After substituting the domain, the entire set of all values possible as outcomes of the dependent variable y.

For e.g. the range of the function F is {1983,1987,1992,1996}. On the other hand, the whole set B is known as the codomain of the function. It is the set which contains all the outputs of the function. So, the set of real numbers is a codomain for every real-valued function. The codomain of the function F is set B.

How to Find the Range of a Function

  • The spread of all the y values from minimum to maximum is the range of the function.
  • In the given expression of y, substitute all the values of x to check whether it is positive, negative or equal to other values.
  • Find the minimum and maximum values for y.
  • Then draw a graph for the same.

 

Till now, we have represented functions with upper case letters but they are generally represented by lower case letters. If f is a function from set A to B and (a,b) ∈ f, then f(a) = b. b is called the image of a under f and a is called pre-image of b under f.

Summary

  • Domain is defined as the entire set of values possible for independent variables.
  • The Range is found after substituting the possible x- values to find the resulting y-values.

There are several types of functions and some of them even have got funny names e.g. floor function, ceiling function, etc. To know more, visit www.byjus.com and experience fun in learning.

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