Formal Definition of a Function

The informal idea of a function as a "rule" is intuitive but imprecise. This section gives the rigorous, set-theoretic definition of a function based on ordered pairs, removing all ambiguity and providing a foundation for advanced mathematics.

Quick Reference

Concept	Formal Meaning
Ordered pair	$(a, b)$ where $a$ is the first member and $b$ is the second
Function $f$	A set of ordered pairs $(x, y)$ with no two pairs sharing the same first member
Domain	Set of all first members of pairs in $f$
Range	Set of all second members of pairs in $f$
Function notation	$y = f (x)$ means the pair $(x, y)$ belongs to $f$

Why a Formal Definition?

Previously, we described a function as a "rule" or "procedure" that assigns to each element of a set $A$ exactly one element in a set $B$ . However, the words rule, procedure, and assigns are not mathematical concepts: they can mean different things to different people. To make the definition completely precise, we reformulate it using the mathematical concept of an ordered pair.

Ordered Pairs

An ordered pair $(a, b)$ consists of two objects, where $a$ is the first member and $b$ is the second. Two ordered pairs $(a, b)$ and $(c, d)$ are equal if and only if both members match:

(a, b) = (c, d) \Leftrightarrow a = c and b = d .

You have already seen ordered pairs in coordinate geometry: the point $(3, - 4)$ is different from the point $(- 4, 3)$ . In an ordered pair, order matters.

Definition of a Function

A function $f$ is a set of ordered pairs $(x, y)$ in which no two pairs share the same first member.

The set of all elements $x$ that appear as first members of pairs in $f$ is called the domain of $f$ . The set of all second members $y$ is called the range (or set of values) of $f$ .

Think of a function as a two-column table: the left column holds $x$ -values (the domain), the right column holds the corresponding $y$ -values (the range). The key rule is that no two rows can have the same $x$ -value with different $y$ -values. In set notation:

for every (x, y) \in f and (x, z) \in f \Rightarrow y = z .

Let

f = {(2, - 1), (1, 7), (3, 5), (5, - 1)} .

Solution

f

is a function because no two pairs share the same first element. The domain and range are:

Notice that the pairs

(2, - 1)

and

(5, - 1)

share the same second element

- 1

. That is perfectly fine: different inputs may produce the same output. What is forbidden is the same input producing two different outputs. Now consider

g = {(2, - 1), (2, 0), (1, 7), (3, 5), (5, 8)} .

g

is not a function because both

(2, - 1)

and $(2, 0)$ belong to

g

: the input $2$ is mapped to two different outputs

- 1

and $0$.

Function Notation Revisited

Since for every $x$ in the domain there is precisely one $y$ with $(x, y) \in f$ , once $x$ is specified, $y$ is uniquely determined. This justifies the familiar notation

y = f (x),

which simply means "the pair $(x, y)$ belongs to the set $f$ ." The notation is more efficient and readable than writing $(x, y) \in f$ every time.

Domain, Range, and Codomain

When we write $f : A \to B$ , we declare:

$A$ is the domain of $f$ : the set of allowable inputs.
$B$ is the codomain of $f$ : the set that outputs are supposed to lie in.
The range of $f$ is the set of outputs actually produced: $Rng (f) = {f (x) ∣ x \in A}$ , which is always a subset of $B$ .

The range may be strictly smaller than the codomain. For example, $f : ℝ \to ℝ$ defined by $f (x) = x^{2}$ has codomain $ℝ$ but range $[0, \infty)$ .

In the next section, we will study domain and range of a function in more detail and various examples.

Frequently Asked Questions

Why define a function as a set of ordered pairs instead of a rule?

Defining a function as a set of ordered pairs is precise and unambiguous. The word "rule" is informal: two different-looking rules (e.g.,

f (x) = x + 2

and

g (x) = \frac{x^{2} - 4}{x - 2}

for

x \neq 2

) might define the same or different functions depending on their domains. The ordered-pair definition settles such questions immediately.

Is the pair $(2, 5)$ in the same function as $(5, 2)$?

There is no contradiction: a function can contain both $(2, 5)$ and $(5, 2)$ as long as those are the only pairs with first members 2 and 5, respectively. The function that contains both pairs would satisfy

f (2) = 5

and

f (5) = 2

Can the range equal the codomain?

Yes. When the range equals the codomain, the function is called surjective (or onto). For example,

f : ℝ \to ℝ

f (x) = x^{3}

has range

ℝ

, which equals its codomain.

Is a function the same as a relation?

A relation from

A

B

is any set of ordered pairs with first members in

A

and second members in

B

. A function is a special kind of relation where no two pairs share the same first member. Every function is a relation, but not every relation is a function.