Find domain and range of the relation R = {(1,2),(2,4),(3,6)}

Question

Find the domain and range of the relation $R = \{(1, 2), (2, 4), (3, 6)\}$ .

Solution — Step by Step

A relation $R$ is a set of ordered pairs $(x, y)$ .

The domain of $R$ is the set of all first elements (x-values) in the ordered pairs.
The range of $R$ is the set of all second elements (y-values) in the ordered pairs.

From $R = \{(1, 2), (2, 4), (3, 6)\}$ :

First elements: $1, 2, 3$

\text{Domain of } R = \{1, 2, 3\}

Second elements: $2, 4, 6$

\text{Range of } R = \{2, 4, 6\}

Notice that every $y = 2x$ in this relation. The domain $\{1, 2, 3\}$ maps to the range $\{2, 4, 6\}$ via the rule $y = 2x$ .

This relation is also a function because each element of the domain maps to exactly one element of the range (no repeated first elements).

Why This Works

The domain collects all inputs (x-values) and the range collects all outputs (y-values). Think of domain as “what values can go in” and range as “what values come out.”

The codomain (not asked here) would be the entire set from which the range values are drawn — if we said $R: \{1,2,3\} \to \mathbb{N}$ , then codomain = $\mathbb{N}$ but range = $\{2, 4, 6\}$ .

Common Mistake

Students sometimes swap domain and range — listing $\{2, 4, 6\}$ as the domain. Remember: domain = first (left) elements, range = second (right) elements. In the ordered pair $(a, b)$ , $a$ is always the domain element and $b$ is the range element.

If the question asks whether this relation is a function: check if any x-value repeats. Here, 1, 2, and 3 each appear exactly once as first elements, so yes — $R$ is a function. If the set were $\{(1,2), (1,4), (3,6)\}$ , the value 1 maps to both 2 and 4, making it NOT a function.

Question

Solution — Step by Step

Why This Works

Common Mistake

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