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State The Domain and Range for The Following Relation Calculator

Reviewed by Calculator Editorial Team

Determining the domain and range of a relation is a fundamental concept in mathematics. This guide explains how to find these essential components of a relation and provides a calculator to help you through the process.

What is Domain and Range?

The domain of a relation is the set of all possible input values (x-values) for which the relation is defined. The range is the set of all possible output values (y-values) that the relation can produce.

For a function, the domain is all real numbers unless restricted by the function's definition. The range is all possible outputs based on the domain.

Note: For relations that are not functions, the domain and range can be more complex as multiple outputs may correspond to a single input.

How to Find the Domain

To find the domain of a relation:

  1. Identify all the input values (x-values) for which the relation is defined.
  2. If the relation is given as a set of ordered pairs, the domain is the set of all first elements of the pairs.
  3. For a function, consider any restrictions such as square roots of negative numbers or division by zero.

Domain = {x | (x, y) ∈ R}

How to Find the Range

To find the range of a relation:

  1. Identify all the output values (y-values) that the relation produces.
  2. If the relation is given as a set of ordered pairs, the range is the set of all second elements of the pairs.
  3. For a function, consider the possible outputs based on the domain.

Range = {y | (x, y) ∈ R}

Examples

Example 1: Ordered Pairs

Given the relation R = {(1, 2), (2, 3), (3, 4)}:

  • Domain = {1, 2, 3}
  • Range = {2, 3, 4}

Example 2: Function

For the function f(x) = √(x - 1):

  • Domain = {x | x ≥ 1}
  • Range = {y | y ≥ 0}

FAQ

What is the difference between domain and range?
The domain is the set of all possible input values, while the range is the set of all possible output values.
How do I find the domain of a function?
Identify any restrictions in the function's definition, such as square roots of negative numbers or division by zero.
Can the domain and range be the same?
Yes, if the relation maps every input to itself, such as the identity function f(x) = x.