Domain and Range of a Relation

Understanding the Domain and Range of a Relation is a key part of learning mathematics. The domain is all the possible input values, while the range is all the possible output values. Knowing how to find these can help you understand functions and graphs better.

In this guide, we will explain what the domain and range are, how to find them, and provide examples. We will also look at specific cases like the domain and range of trigonometric functions. Whether you're a student or just someone curious about math, this guide will make these concepts clear and easy to understand.

1.0Relation

A Relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. This subset is formed by establishing a relationship between the first element and the second element of the ordered pairs in A × B. The set of all first elements in the relation R is referred to as the domain of R. The set of all second elements, also known as images, is called the range of R.

2.0Domain of a Relation

The Domain of a Relation is the set of all possible input values. These inputs are the first elements in the ordered pairs of the relation.

Example: Consider the relation R = {(1, 4), (2, 5), (3, 6)}.

Domain: {1, 2, 3}

3.0Range of a Relation

The Range of a Relation is the set of all possible output values. These outputs are the second elements in the ordered pairs of the relation.

Example: Using the same relation R = {(1, 4), (2, 5), (3, 6)}

Range: {4, 5, 6}

4.0How Do You Find the Domain and Range of a Relation?

Domain of a given Relation

To find the Domain of a Relation, list all the first elements of the ordered pairs in a relation and remove any duplicates. The resulting set is the domain.

Range of a given Relation

To find the Range of a Relation, list all the second elements of the ordered pairs in the relation and remove any duplicates. The resulting set is the range.

5.0Domain and Range of Trigonometric Functions

6.0Domain and Range Relations Solved Examples

Example 1: Let A = {1, 2, 3, 4} and B = {x, y, z}. Consider the subset R = {(1, x), (1, y), (2, z), (3, x)} of A × B. Is R a relation from A to B? If yes, find domain and range of R. Draw arrow diagram of R.

Solution:

Since every subset of A × B is a relation from A to B, therefore, R is also a relation from A to B.

Domain of R = {1, 2, 3} 

Range of R = {x, y, z}

Example 2: If A=$\{1,2,3,4\}$ and B=$\{a,b,c,d\}$, then find the domain and range of the relation R=$\{(1,a),(2,b),(3,a)\}$ from $Ato\text{ B. }$

Solution:

Domain =$\{1,2,3\}$

Range = $\{a,b\}$

Example 3: Find Domain of Relation: R = {(1, 3), (2, 3), (3, 3), (4, 5)} 

Solution: 

Domain of R = {1, 2, 3, 4}

Range of R = {3, 5}

Example 4: Let R = {(a, a³): a is a prime number less than 5}. Find The range of R.

Solution:

R = {(2, 8), (3, 27)}

Domain of R = {2, 3}

Range of R = {8, 27}

Example 5: Let R=$\left\{\left(a,\frac{1}{a}\right):a\in N\text{ and }1<a<5\right\}$. Find the domain and range of R.

Solution:

$R=\left\{\left(2,\frac{1}{2}\right),\left(3,\frac{1}{3}\right),\left(4,\frac{1}{4}\right)\right\}$

Domain of R = (2, 3, 4)

Range of $\mathrm{R}=\left\{\frac{1}{2},\frac{1}{3},\frac{1}{4}\right\}$

Example 6: Find the domain and range of the relation R = {(–1,1), (1,1), (–2,4), (2,4)}.

Solution: 

Domain of R = (–1, 1, –2, 2)

Range of R = {1, 4}

7.0Domain and Range of a Relation Practice Question

1. Let R = {(a, b): a, b ∈ N and b = a + 5, a < 4}. Find the domain and range of R.
2. Let R = {(a, a³): a is a prime number less than 10}. Find (i) R (ii) domain (R) (iii) range (R).
3. Let R = {(x, y): x + 2y = 8} be a relation on N. Write the range of R.
4. Let R= {(a, b): a, b ∈ N and a + 3b = 12}. Find the domain and range of R.
5. Let R= {(a, b): b= |a – 1|, a ∈ Z and |a| ˂ 3}. Find the domain and range of R.