Surjective function

A function $f:X\to Y$ is surjective (or onto) if

Equivalently, $f(X)=Y$ (the image equals the codomain ).

Surjectivity depends on the specified codomain $Y$, not just on the rule $x\mapsto f(x)$. Many constructions in analysis naturally produce surjections (e.g., quotient maps, parameterizations) and surjectivity is required for an inverse to be defined on all of $Y$.

Examples:

• $\sin:\mathbb{R}\to[-1,1]$ is surjective.
• The same rule $\sin:\mathbb{R}\to\mathbb{R}$ is not surjective since no real $x$ satisfies $\sin x = 2$.
• $f:\mathbb{R}\to[0,\infty)$, $f(x)=x^2$ is surjective.