Surjective Function

• Definition: A function $f:A\to B$ is surjective if every element of $B$ has at least one preimage in $A$.
• Formally:

$$\forall y\in B,\exists x\in A\mathrm{such}\mathrm{that}f(x)=y$$

Translation: “For every element $y$ in the codomain $B$, there exists at least one element $x$ in the domain $A$ such that $f(x)=y.$”

• Meaning: The function “covers” the entire codomain.
• Example: $f(x)=x^3$ from $\mathbb{R}\to \mathbb{R}$ is surjective.