surjective

Surjective (onto) function

A function $f:A\to B$ is called surjective if $\forall y\in B\exists x\in A:f(x)=y$
In words: every element in the codomain is the image of at least one element in the domain.

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EXAMPLE

$f_1:\mathbb{R}\to {\mathbb{R}}_0^{+}:x\mapsto x^2$ is not injective, because $f_1(-1)=f_1(1)$ and it is not surjective, because $f_1(x)\ge 0\forall x\in \mathbb{R}$ (negative real numbers are not in the codomain).
$f_2:{\mathbb{R}}_0^{+}\to \mathbb{R},x\mapsto x^2$ is injective, but not surjective because we canot reach negative numbers.
$f_3:\mathbb{R}\to {\mathbb{R}}_0^{+},x\mapsto x^2$ is not injective but surjective ($x^2=z$ has a solution for all $z\in {\mathbb{R}}_0^{+}$).
$f_4:{\mathbb{R}}_0^{+}\to {\mathbb{R}}_0^{+},x\mapsto x^2$ is bijective.

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