It's in my hands to have a surjective function Let $f$ be any function $A \to B$.
By definition $f$ is a surjective function if $\space \forall y \in B \space \exists \space x \in A \space( \space f(x)=y \space)$.
So, for any function I only have to ensure that there doesn't "remain" any element "alone" in the set $B$. In other words, the range set of the function has to be equal to the codomain set.
The range depends on the function, but the codomain can be choose by me. So if I chose a codomain equal to the range I get a surjective function, regardless the function that is given.
M'I right? 

There are intrinsic properties and extrinsic properties. Being surjective is an extrinsic property. If you are not given a particular codomain you cannot conclude whether or not a function is surjective. 
Being injective, on the other hand, is an intrinsic property. It depends only on the function as a set of ordered pairs. 
