Domain, Codomain, Range

The Three Parts of a Function

Every function $f: A \to B$ has three important sets:

• Domain — the set of inputs
• Codomain — the set of possible outputs
• Range — the set of outputs that actually get used

Definitions

$f: A \to B$

Domain = $A$ = all inputs

Everything that goes into the function.

Codomain = $B$ = all possible outputs

Everything that could come out.

Range = outputs that actually occur

Everything that does come out.

Example

$f: \{1, 2, 3\} \to \{a, b, c, d\}$

$f = \{(1, a), (2, a), (3, b)\}$

Notice: $c$ and $d$ are in the codomain but nothing maps to them.

Range vs Codomain

The range is always a subset of the codomain:

$\text{Range} \subseteq \text{Codomain}$

Sometimes they’re equal — every possible output gets used.

Sometimes the range is smaller — some outputs go unused.

Another Example

$f(x) = x^2 \text{ where } f: \mathbb{R} \to \mathbb{R}$

• Domain = $\mathbb{R}$ (all real numbers can be squared)
• Codomain = $\mathbb{R}$ (we declared outputs are real numbers)
• Range = $\mathbb{R} \geq 0$ (only non-negative numbers come out)

Why? Squaring never gives a negative result. So $-5$ is in the codomain, but no input maps to it.