1. What the Codomain Means
The codomain of a function is the set that contains all possible outputs the function is allowed to produce. It is chosen while defining the function — not after seeing what outputs actually appear.
The codomain is about what is permitted, not what is actually produced.
2. Difference Between Codomain and Range
This is a key idea:
- Codomain is the set we declare as the target set.
- Range is the set of outputs that the function actually produces.
2.1. Relationship
\text{Range}(f) \subseteq \text{Codomain}(f)
3. Formal Definition
If a function \( f: A \to B \) is defined, then B is the codomain. This is fixed in the function's definition.
3.1. Definition
\text{Codomain}(f) = B
4. Example to See the Difference Clearly
Let a function be defined as:
f : \mathbb{R} \to \mathbb{R}
f(x) = x^2
4.1. Codomain
\text{Codomain} = \mathbb{R}
This means all real numbers are allowed as possible outputs.
4.2. Range
\text{Range} = [0, \infty)
The function actually produces only non-negative numbers.
5. Codomain in Set of Ordered Pairs
If a function is listed as ordered pairs, the codomain is usually mentioned separately. It does not have to match the actual set of second elements.
5.1. Example
Let:
A = \{1,2,3\}, \; B = \{4,5,6,7\}
f = \{(1,4),(2,4),(3,5)\}
Here B is the codomain, even though 6 and 7 never appear as outputs.
6. Codomain Depends on How the Function Is Defined
Two functions can have the same formula but different codomains if defined differently.
6.1. Example
f_1 : \mathbb{R} \to \mathbb{R}, \; f_1(x) = x^2
f_2 : \mathbb{R} \to [0, \infty), \; f_2(x) = x^2
Both use the same rule x², but f1 can have negative outputs allowed (even though not produced), while f2 restricts possible outputs to non-negative values.
7. Why Codomain Matters
The choice of codomain affects how we classify a function. For example, whether a function is “onto” depends on whether its range fills the entire codomain. Codomain also matters when defining inverses or combining functions.