Relations and Functions — Types, Domain & Range

The one-line answer: a relation is any set of ordered pairs, and a function is a relation where each input has exactly one output. To find the domain, collect the first coordinates; to find the range, collect the outputs that actually appear. Once the one-input-one-output rule is clear, domain, range, and mapping type fall into place.

Relation vs. function side by side

The relation $R = \{(1,2),(1,3),(2,3)\}$ is not a function: the input $1$ is paired with both $2$ and $3$. The set $f = \{(1,2),(2,3),(3,3),(4,5)\}$ is a function, because no first coordinate is paired with two different second coordinates. Different inputs sharing the same output is fine.

How to find domain and range

The domain is the set of all inputs, so it comes from the first coordinates. The range is the set of outputs that actually appear, so it comes from the second coordinates. Using

we get

Notice $3$ appears twice as an output but is written once: the range lists distinct outputs, not how many times they occur. If a problem also gives a codomain, do not treat it as the range. The codomain is the larger target set outputs are allowed to come from; the range is the subset the function actually hits.

Which mapping types can be functions

When people classify relations and functions, they usually mean one of these patterns:

Only the first two can be functions. A one-to-many relation is never a function, because one input would have multiple outputs.

Worked example: classify in one pass

Let

Writing out the pairs:

The domain is all first coordinates:

The range is all outputs that occur:

Is it a function? Yes — each input appears once with exactly one output. What type? Many-to-one, not one-to-one, because both $-2$ and $2$ map to $4$, and both $-1$ and $1$ map to $1$. The point students miss: repeated outputs do not break a function; repeated inputs with different outputs do.

Reading it from a graph

If a relation is shown as a graph, the vertical line test is the quick check. If some vertical line meets the graph at more than one point, one $x$-value has more than one $y$-value, so the graph is not a function. This works only because the graph is read as pairs $(x,y)$ — it is a visual restatement of the one-input-one-output rule.

Common confusion points

Thinking repeated outputs break a function

They do not. A function can be many-to-one. The breaker is repeated inputs with different outputs.

Mixing up range and codomain

If the codomain is, say, $\{0,1,2,3,4,5\}$, the range might still be only $\{0,1,4\}$. Range means actual outputs, not all allowed outputs.

Forgetting domain restrictions

A formula does not always tell the whole story. For example, $f(x)=1/x$ is undefined at $x=0$, so $0$ cannot be in the domain.

Assuming every relation is a function

Relations are the broader idea; functions are the stricter case inside that category.

Where each idea is used

Relations describe which objects connect to which others — set theory, databases, graph theory, coordinate geometry. Functions are even more central: algebra, calculus, statistics, physics, and computer science all use them to describe how one quantity depends on another. Whenever you see "input this value, get that output," you are usually looking at a function.

Try the comparison yourself

Build a small relation on the domain $\{1,2,3\}$. First make one that is not a function by giving one input two different outputs. Then change a single pair so it becomes a function, and compare the domain and range before and after. Seeing the same set flip from relation to function in one edit is the fastest way to lock in the distinction.