How to identify the domain and range of a relation from ordered pairs

Key takeaway: When you work with a relation, the input values and output values play different roles. The domain is the set of inputs, and the range is the set of outputs that actually appear. That sounds simple, but it helps to separate the rule from the list of pairs.

A relation is fully described by two things:

1. the rule that connects inputs to outputs,
2. the domain of allowed input values.

For the relation

$\{(x,y): y=x+1,\ x\in\{1,2,3,4\}\},$

the rule is $y=x+1$ and the domain is $\{1,2,3,4\}$.

Now substitute each value of $x$:

• if $x=1$, then $y=2$
• if $x=2$, then $y=3$
• if $x=3$, then $y=4$
• if $x=4$, then $y=5$

So the relation is

$\{(1,2),(2,3),(3,4),(4,5)\}.$

• Domain: $\{1,2,3,4\}$
• Range: $\{2,3,4,5\}$

For the examples you listed:

a.

If the domain is $S=\{1,3,5\}$ and the range is $\{1,2,4,6\}$, then the relation must map each input in $S$ to one or more outputs in that range, depending on the rule.

b.

If the domain is $T=\{-3,4,5,7\}$ and the range is $\{5,12,-6\}$, then again the exact pairs depend on the rule, but the domain still means the allowed inputs and the range means the outputs that occur.

The main idea is: domain = what goes in, range = what comes out.

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Pitfalls the pros know 👇 Students often mix up range with codomain or with the set of all possible outputs suggested by the rule. Here, the range is only the outputs that actually show up from the listed domain. Another easy mistake is forgetting that the same output can appear more than once in the ordered pairs; sets do not repeat elements.

What if the problem changes? If the domain changes, the relation changes too, even when the formula stays the same. For example, with $y=x+1$:

• if the domain is $\{1,2,3\}$, the range is $\{2,3,4\}$
• if the domain is $\{0,2,4\}$, the range is $\{1,3,5\}$

Same rule, different domain, different relation.

Tags: domain, range, ordered pairs