The domain of a function f ( x ) is the set of all values for which the function is defined, and the range of the function is the set of all values that f takes.
(In grammar school, you probably called the domain the replacement set and the range the solution set. They may also have been called the input and output of the function.)
Example 1:
Consider the function shown in the diagram.
Here, the domain is the set { A , B , C , E } . D is not in the domain, since the function is not defined for D .
The range is the set { 1 , 3 , 4 } . 2 is not in the range, since there is no letter in the domain that gets mapped to 2 .
You can also talk about the domain of a relation , where one element in the domain may get mapped to more than one element in the range.
Example 2:
Consider the relation { ( 0 , 7 ) , ( 0 , 8 ) , ( 1 , 7 ) , ( 1 , 8 ) , ( 1 , 9 ) , ( 2 , 10 ) } .
Here, the relation is given as a set of ordered pairs. The domain is the set of x -coordinates, { 0 , 1 , 2 } , and the range is the set of y -coordinates, { 7 , 8 , 9 , 10 } . Note that the domain elements 1 and 2 are associated with more than one range elements, so this is not a function.
But, more commonly, and especially when dealing with graphs on the coordinate plane, we are concerned with functions, where each element of the domain is associated with one element of the range. (See The Vertical Line Test .)
Example 3:
The domain of the function
f ( x ) = 1 x
is all real numbers except zero (since at x = 0 , the function is undefined: division by zero is not allowed!).
The range is also all real numbers except zero. You can see that there is some point on the curve for every y -value except y = 0 .
Domains can also be explicitly specified, if there are values for which the function could be defined, but which we don't want to consider for some reason.
Example 4:
The following notation shows that the domain of the function is restricted to the interval ( − 1 , 1 ) .
f ( x ) = x 2 , − 1 < x < 1
The graph of this function is as shown. Note the open circles, which show that the function is not defined at x = − 1 and x = 1 . The y -values range from 0 up to 1 (including 0 , but not including 1 ). So the range of the function is
0 ≤ y < 1 .
Before learning about Domain and Range of a Relation firstly know what Relations are. A relation is a rule that relates an element from one set to the other set. Consider two non-empty sets A and B then the relation is a subset of Cartesian Product AxB. The domain is the set of all first elements of the ordered pairs. The range on the other hand is the set of all second elements of the ordered pairs. However, Range
includes only the elements used by the function. There lies in a trick in the range i.e. Set B can be equal to the range of relation or bigger than that. This is because there can be elements in Set B that aren’t related to Set A. If R be a relation from Set A to Set B then the Set of first elements belonging to the ordered pair is called the Domain of R. We can represent the Domain as such Dom(R) = {a ∈ A: (a, b)
∈ R for some b ∈ B} The Set of Second Components belonging to the ordered pair is called the Range of R. It can be denoted as follows R = {b ∈ B: (a, b) ∈R for some a ∈ A} Thus, Domain and Range are given by Domain (R) = {a : (a, b) ∈ R} and Range (R) = {b : (a, b) ∈ R}. 1. State the domain and range of the following relation: (eye color, student’s name).Domain and Range Definition
Solved Examples on Domain and Range of a Relation
A = {(blue, John), (green, William), (brown,Wilson),
(blue, Moy), (brown, Abraham), (green, Dutt)}. State whether the relation is a function?
Solution:
Domain: {blue, green, brown} Range: {John, William, Wilson, Moy, Abraham, Dutt}
No, the relation is not a function since the eye colors are repeated.
2. State the domain and range of the following relation: {(4,3), (-1,7), (2,-3), (7,5), (6,-2)}?
Solution:
The domain is the first component of the ordered pairs. Whereas, Range is the Second Component of the ordered pairs. Remove the duplicates if any are present.
Domain = {4, -1, 2, 7, 6} Range = {3, 7, -3, 5, -2}
3. From the following Arrow Diagram find the Domain and Range and depict the relation between them?
Solution:
Domain = {3, 4, 5}
Range = {3, 4, 5, 6}
R = {(3, 4), (4, 6), (5, 3), (5, 5)}
4. Determine the domain and range of the relation R defined by
R = {x – 2, 2x + 3} : x ∈ {0, 1, 2, 3, 4, 5}
Solution:
Given x = {0, 1, 2, 3, 4, 5}
x = 0 ⇒ x – 2 = 0 – 2 = -2 and 2x + 3 = 2*0 + 3 = 3
x = 1 ⇒ x-2 = 1-2 = -1 and 2x+3 = 2*1+3 = 5
x = 2 ⇒ x-2 = 2-2 = 0 and 2x+3 = 2*2+3 = 7
x = 3 ⇒ x-2 = 3-2 = 1 ad 2x+3 = 2*3+3 = 9
x = 4 ⇒ x-2 = 4-2 = 2 and 2x+3 = 2*4+3 = 11
x = 5 ⇒ x-2 = 5-2 =3 and 2x+3 = 2*5+3 = 13
Hence R = {-2, 3), (-1, 5), (0, 7), (1, 9), (2, 11), (3, 13)
Domain of R = {-2, -1, 0, 1, 2, 3}
Range of R = {3, 5, 7, 9, 11, 13}
5. The below figure shows a relation between Set x and Set y. Write the same in Roster Form, Set Builder Form, and find the domain and Range?
Solution:
In the Set Builder Form R = {(x, y): x is the square of y, x ∈ X, y ∈ Y}
In Roster Form R = {(2, 1)(4, 2)}
Domain = {2, 4}
Range = {1, 2}
6. The Arrow Diagram Shows the Relation R from Set C to Set D. Write the relation R in Roster Form?
Solution:
We observe the relation R using the Arrow Diagram Above
From that Relation R in Roster Form = {(2,20) ; (2, 40) ; (4, 40) ; (3, 30)}