How do you prove that a function is surjective?

[sara_87]

how do you prove that a function is surjective ?

i know that surjective means it is an onto function, and (i think) surjective functions have an equal range and codomain?

 

[Hurkyl]

There are lots of ways one might go about doing it. The most direct is to prove every element in the codomain has at least one preimage. i.e. for a function $f:X \to Y$, to show

$$\forall y \in Y :\exists x \in X: f(x) = y$$

 

[sara_87]

how can i prove if f(x)= x^3, where the domain and the codomain are both the set of all integers: Z, is surjective or otherwise...the thing is, when i do the prove it comes out to be surjective but my teacher said that it isn't.

this is what i did:

y=x^3

and i said that that y belongs to Z and x^3 belong to Z so it is surjective

this is obviously wrong, but i don't know what I'm doing wrong!

 

[Hurkyl]

Because, to repeat what I said, you need to show for every y, there exists an x such that f(x) = y!


You claim f is surjective -- that means (for example) that you can find an x such that f(x) = 2.

 

[sara_87]

'Because, to repeat what I said, you need to show for every y, there exists an x such that f(x) = y!'
okay, easy! lol
i read that ten thousand times already! just give it time to sink in...okay it has sunk in

i guess it is not surjective then...thanx for opening up my eyes

 

[HallsofIvy]

Does there exist x in Z such that, for example, f(x)= x³= 2?