# How to Find Domain and Range of Relations

Domain and Range are two basic concepts in the definition of Relations, and by changing them, the definition of Relations also changes. In this article, this topic is explained in detail and a comprehensive definition of Domain and Range is provided.

**A step-by-step guide to Find Domain and Range of Relations **

All the input values (\(x\)-values) accepted by the relation form the domain of the relation. The range of a relation is all the output values (\(y\)-values) produced by the relation.

Note that some relations have limitations in their domain or range, such as an equation that is defined only for negative numbers or only for positive numbers. For these cases, consider the limitations when determining the domain and range.

For relations expressed as a set of \((x, y)\) pairs, the range is the set of all values of \(x\) and the range is the set of all values of \(y\).

Also, in a certain function in the form of \(y = f(x)\), the domain is a set of all values of \(x\) for which \(f(x)\) is defined, and the range includes the set of all values of \(y\) resulting from the output of placing x in the function relationship.

The domain of a graph is the set of all \(x\)-values of points on the graph, while the range of the graph is the set of all \(y\)-values of points on the graph.

If the graph is a linear relationship that extends to infinity in both directions, the range will be all real numbers, while the range is determined by the \(y\)-intercept and slope of the line.

**Finding Domain and Range of Relations-Example 1**

What is the domain and range of the following relation:

{\((4, 7), (-3, 5), (6, 1), (0, 3), (10, 2)\)}

**Solution:** The domain contains \(x\)-values of a relation and the range includes \(y\)-values of a relationship. So, Domain\(=\){\(4, -3, 6, 0, 10\)} and Range\(=\){\(7, 5, 1, 3, 2\)}

**Finding Domain and Range of Relations-Example **2

Find the domain and range of the following relation:

\(R=\){\(5x-2, 2x+1\)} : \(x∈\){\(1, 2, 3, 4\)}

**Solution:** In the question, we have given \(x∈\){\(1, 2, 3, 4\)}. So, put these values in two equations of \(R=\){\(5x-2, 2x+1\)} to find the domain and range of the relation:

\(x=1\)→\(5x-2=5(1)-2=3, 2x+1=2(1)+1=3\)

\(x=2\)→\(5x-2=5(2)-2=8, 2x+1=2(2)+1=5\)

\(x=3\)→\(5x-2=5(3)-2=13, 2x+1=2(3)+1=7\)

\(x=4\)→\(5x-2=5(4)-2=18, 2x+1=2(4)+1=9\)

\(R=\)={\((3, 3), (8, 5), (13, 7), (18, 9)\)}

The domain of \(R=\){\(3, 8, 13, 18\)} and the range of \(R=\){\(3, 5, 7, 9\)}.

**Exercises for** **Finding Domain and Range of Relations**

1-What is the domain and range of the following relation:

{\((0, 3), (2, 4), (4, 6), (6, 9), (8, 15), (10, 20)\)}

2-Find the domain and range of the following relation:

\(R=\){\(x-3, 3x+2\)} : \(x∈\){\(-1, 0, 1, 3\)}

1-Domain\(=\){\(0, 2, 4, 6, 8, 10\)} , Range\(=\){\(3, 4, 6, 9, 15, 20\)}

2-Domain\(=\){\(-4, -3, -2, 0\)} , Range\(=\){\(-1, 2, 5, 11\)}

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