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Domain and Range Relations are part of sets. The set which consists of all the first elements of the ordered pairs of a relation, R, is called the Domain of the Relation. However, the set containing all the second elements is called the range of the relation.
- Relation in mathematics, just like a Cartesian product, will have ordered pairs.
- The second element in these ordered pairings is the copy of the first element, while the first element is the pre-image of the second set of elements.
- This means that one set will have all the pre-images, while the other will contain all the images.
- The relation domain is the set that has all the initial elements of all the ordered pairs of relations.
- The domain set can be both identical or unidentical to set R.
- The range of the relation is the set that comprises all the second elements.
Also Check: Correlation Coefficient Formula
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Key Terms: Relations, Functions, Range, Domains, Sets, Arrow diagram, Roster, Co-domain, Square Root, Absolute Value Function, Ordered Pairs
What is a Relation?
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Relation can be defined as the dependency between sets of any ordered pairs. The ordered pair group is defined as the relationship between two sets in which the object from each set forms the ordered pair. A relation R is the subset of the cartesian product of any two non-void sets A and B. According to Relation, every element of one set is related to one or more elements of the other set.
- Every input has one or many outputs.
- If (a, b) ∈ R it can be said that a is related to b under the relation R and is represented as aRb.
- An arrow symbol shows the relationship R: A→B between sets A and B.
- Ordered pairs may be finite or infinite in a relation.
- The term “relation on A” expresses a relationship between sets A and A.
- 2mn is the maximum number of relations defined from sets A (with m elements) and (with n elements).
What is a Function?
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A function can be defined as a set of arranged pairs wherein each input component contains only ONE output component related to it. A function may consist of two input values assigned to the same output value but it will not consist of two output values assigned to the same input value.
Interval and Notation
The Interval can be expressed as subsets that can be found on the number line.
Example: All real numbers in range of 1 and 6 consists of intervals such as, 2.5, 3, 4.567, until 6.
Interval notation can be defined as a method of making subsets of the real number line.
Example: (0, 20], [1, 2], (4, 7), [4, 8).
Types of Interval
There are two types of Intervals on a number line:
- Open interval: It does not include the endpoints in the set. The ( ) brackets denote an open interval. Example: (4, 6), (6, 9), (-3, 2).
- Closed interval: It includes the endpoints in the set. It can be denoted by [ ] brackets. Example: [3, 5], [-4, 0].
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Domain and Range of a Relation
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Relation, like Cartesian products, is going to have ordered pairs. And in the ordered pairs, the second element is the image of the first element, while the first element is the preimage of the second.
- Domain: All of the values that go into a relation or a function are known as the domain. All input values (independent values) form the Domain set.
- Range: All of the entities (output) which come out from a relation or a function are called the range. All output values (dependent values) form the Range set.
They can be represented in three different ways, such as:
- Set-builder method: A= {(x, y): y=x+2; x, y ∈N}
- Roster method: A= {(9,11), (10,12), (11,13)}
- Arrow diagram: The arrow representation can be shown as,
Arrow Representation
Also read: NCERT Solutions Class 12 Mathematics Relations and Functions
Domain and Range on a Graph
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In a graph, the x-coordinates (abscissa) represent the domain and the y-coordinates (ordinates) represent the range.
The abscissa represents the domain value, when put into the function, the value that comes as output will lie on the y-axis.
Example: In the below figure, the line y = 1 is a function where all values of x belong to the domain set and for every value of x, there is only 1 as an answer. So, 1 is the range.
Domain values are abscissa and since f is a function of x so, the values of f (ordinates) obtained by putting values of abscissa will make our range. It is easier if we draw a graph and find the range (output) of the function, rather than find values manually.
Since the output is a positive value for the function including 0 and excluding 1. From the graph, it can be concluded that the range of the function in the interval form is [0, 1).
Relations and Functions Detailed Explanation Video
Domain of a Function
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A domain of a function generally refers to "all the values" which go into a function. The domain of a function can be defined as the set of all possible inputs for the function. The general formulas that are widely used to find the domain of types of functions include:
- The domain of a polynomial (including quadratic, linear, cubic, etc) function is R.
- The domain of a square root function √x is x≥0.
- To determine the domain of a rational function y = f(x), the denominator is set as ≠ 0.
- The domain of an exponential function is R.
- The domain of a logarithmic function is x>0.
Range of a Function
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The range of a function can be defined as the set of all its outputs. The general formulas which are used to determine the range of different types of functions are:
- The range of a linear function is R.
- A Quadratic function’s range y = a(x-h)2 + k is: y≥k, if a>0 and y≤k, if a<0
- The range of a square root function is y ≥ 0.
- The range of an exponential function is y>0.
- A Logarithmic function’s Range is R.
- To determine the range of a rational function y = f(x), first, solve x and further consider the denominator ≠ 0.
Relation Representation
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Apart from set notation, there are various ways to represent the relation, such as using tables, plotting it on an XY–axis, or using a mapping diagram.
Relation Representation
Domain and Range Calculation
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If R is a relation from set A to set B in the domain and range of a relation, then
The domain of R is defined as the set of all first components of R's ordered pairs.
Dom(R) = {a ∈ A: (a, b) ∈ R for some b ∈ B}.
The range of R is the set of all second components of ordered pairs that belong to R.
Range(R) = {b ∈ B: (a, b) ∈ R for some a ∈ A}.
Thus, Domain (R) = {a : (a, b) ∈ R} and Range (R) = {b : (a, b) ∈ R}
Domain and Range of Exponential Functions
The function y = ax, a ≥ 0 can be expressed for all real numbers. Thus, the exponential function’s domain is the complete real line. The exponential function is known to give a positive value. Hence, the exponential function’s range is in the form y= |ax+b| is y ∈ R , {y > 0}. Domain = R, Range = (0, ∞).
- Domain: Set R is the domain of the function.
- Range: The exponential function always gives positive real values.
Domain and Range of Absolute Value Function
The function y=|ax+b| can be expressed for all the real numbers. This is to say, the domain of absolute value function is also the set of all real numbers. The absolute value of a number always produces a non-negative value. Hence, the absolute value function’s range of the form y = |ax+b| is y ∈ R | y ≥ 0. Hence, it can be said that:
- Domain = R
- Range = [0, ∞)
Domain and Range of a Square Root Function
The function y= √(ax+b) can be expressed for x ≥ -b/a. Thus, the square root function’s domain is the set of all real numbers greater than or equal to -b/a. Square root is known to result in a non-negative value always. This is why the range of a square root function consists of all non-negative real number sets. Hence,
- Domain = [-b/a,∞)
- Range = [0,∞)
Also read:
Domain and Range of Trigonometric Functions
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The domain and range of the various trigonometric functions include:
| Trigonometric Functions | Domain of Trigonometric Functions | Range of Trigonometric Functions |
|---|---|---|
| Sin θ | (-∞, + ∞) | [-1, +1] |
| Cos θ | (-∞ + ∞) | [-1, +1] |
| Tan θ | R – (2n + 1)π/2 | (-∞, +∞) |
| Cot θ | R – nπ | (-∞, +∞) |
| Sec θ | R – (2n + 1)π/2 | (-∞, -1] U [+1, +∞) |
| Cosec θ | R – nπ | (-∞, -1] U [+1, +∞) |
Read Also: Sin Cos Formulas
Things to Remember
- A function is a relationship in which each input has just one output.
- Interval notation is a way of making subsets of the real number line.
- A domain of a function typically indicates "all the values" which go into a function.
- The range of a function expresses the set of all its outputs.
- A Relation can simply de expressed as the dependency between sets of any ordered pairs.
Previous Year Questions
- If FF is function such that F(0)=2,F(1)=3...
- (x−1)(x2−5x+7)<(x−,1) then x belongs to… [BITSAT 2007]
- Consider the following lists… [AP EAPCET]
- If a+π/2<2tan−1… [KCET 2019]
- The set {x:f(x)=f−1(x)}={1,2}{x:f(x)=f−1(x)}={1,2} f is a bijection and f−1(x)=1+√x−1,x≥1...
- If A={x|x∈N,x≤5},B={x|x∈Z,x2−5x+6=0}… [KCET 2019]
- f : R → R and g : [0, ∞) → R is defined by… [KCET 2019]
- cos[2sin−13/4 + cos−13/4] =… [KCET 2019]
- On the set of positive rationals, a binary operation… [KCET 2019]
Sample Questions
Ques. State the domain and range of the following relation: (eye colour, student’s name).
A = {(blue, John), (green, William), (brown, Wilson), (blue, Moy), (brown, Abraham), (green, Dutt)}. Is the relation a function? (1 mark)
Ans. Domain: {blue, green, brown}
Range: {John, William, Wilson, Moy, Abraham, Dutt}
No, the relation is not a function since the eye colours are repeated.
Ques. If A = {2, 4, 6, 8) B = {5, 7, 1, 9}.
If R be the relation ‘is less than’ from A to B. What is the Domain (R) and Range (R)? (2 marks)
Ans Under this relation (R),
R = {(4, 5); (4, 7); (4, 9); (6, 7); (6, 9), (8, 9) (2, 5) (2, 7) (2, 9)}
Thus, Domain (R) = {2, 4, 6, 8} and Range (R) = {1, 5, 7, 9}
Ques: From the following Arrow Diagram find the Domain and Range and depict the relation between them? (2 marks)
Ans. Domain = {3, 4, 5}
Range = {3, 4, 5, 6}
R = {(3, 4), (4, 6), (5, 3), (5, 5)}
Ques: Let R be the relation ‘is factor of’ from A to B.
(a) Write R in the roster form. Also, find Domain and Range of R.
(b) Draw an arrow diagram to represent the relation. (3 marks)
Ans: (a) Clearly, R consists of elements (a, b) where a is a factor of b.
Therefore, Relation (R) in the roster form is R = {(2, 8); (2, 10); (3, 9); (4, 8), (5, 10)}
Therefore, Domain (R) = Set of all first components of R = {2, 3, 4, 5} and Range (R) = Set of all second components of R = {8, 10, 9}
(b) The arrow diagram:
Ques. The arrow diagram shows the relation (R) from set A to set B. Write this relation in the roster form. (2 marks)
Ans. Clearly, R consists of elements (a, b), such that ‘a’ is square of ‘b’
i.e., a = b2.
So, in roster form R = {(9, 3); (9, -3); (4, 2); (4, -2); (16, 4); (16, -4)}
Ques. Find the domain and range of the relation R defined by R = {x + 2, x + 3}: x ∈ {0, 1, 2, 3, 4, 5} (4 marks)
Ans: Since, x = {0, 1, 2, 3, 4, 5}
Therefore,
x = 0 ⇒ x + 2 = 0 + 2 = 2 and x + 3 = 0 + 3 = 3
x = 1 ⇒ x + 2 = 1 + 2 = 3 and x + 3 = 1 + 3 = 4
x = 2 ⇒ x + 2 = 2 + 2 = 4 and x + 3 = 2 + 3 = 5
x = 3 ⇒ x + 2 = 3 + 2 = 5 and x + 3 = 3 + 3 = 6
x = 4 ⇒ x + 2 = 4 + 2 = 6 and x + 3 = 4 + 3 = 7
x = 5 ⇒ x + 2 = 5 + 2 = 7 and x + 3 = 5 + 3 = 8
Hence, R = {(2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8)}
Therefore, Domain of R = {a: (a, b) ∈R} = Set of first components of all ordered pair belonging to R. Therefore, Domain of R = {2, 3, 4, 5, 6, 7}
Range of R = {b: (a, b) ∈ R} = Set of second components of all ordered pairs belonging to R.
Therefore, Range of R = {3, 4, 5, 6, 7, 8}
Ques. The below figure shows a relation between Set x and Set y. Represent this in Roster Form, Set Builder Form. Additionally, also find the domain and Range. (2 marks)
Ans. 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}
Ques. Let A = {3, 4, 5, 6, 7, 8}. Write a relation R from A to A by R = {(x, y): y = x - 1}.
a) Portray this relation using an arrow diagram.
b) Find the domain and range of R. (2 marks)
Ans: By definition of relation
R = {(4, 3) (5, 4) (6, 5)}
The corresponding arrow diagram is shown.
So, domain = {4, 5, 6} and Range = {3, 4, 5}
Ques. The figure displays a relation between the sets A and B. Write this relation in
(i) Set builder form
(ii) Roster form
(iii) Find the domain and range (3 marks)
Ans: We observe that the relation R is 'a’ is the square of ‘b'.
In set-builder form R = {(a, b): a is the square of b, a ∈ A, b ∈ B}
In roster form R = {(4, 2) (4, -2) (9, 3) (9, -3)}
Therefore, Domain of R = {4, 9}
Range of R = {2, -2, 3, -3}
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