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So i'm studying for my discrete mathematics final, and as I review over surjective/onto functions, i'm having issues explicitly proving functions are onto that don't have a simple inverse to find an plug in. Something like:

$$f(x) = 4x+2$$ is quite easy to prove either onto/not onto because the inverse is easy to find and the internet is littered with examples just like that. But when I come across something like:

$$f(x) = x^3-x$$ or: $$f: \Bbb Z \rightarrow \Bbb Z\ by\ f(x) = 2016x^3-263x$$

finding the inverse can be messy and aggravating. Going thru my notes from the year doesn't help me since our examples never got that difficult(but the exams did) and my prof isn't the most helpful via email.

So, what would be the best way to explicitly prove if these functions are onto? (no induction, please). Thanks!

edit: So here is the proof my professor gave of the second equation, but i'm not sure how it applies outside of this particular problem:

Claim: f(x) is not onto.

We claim that 1 $\in$ range(f).

Suppose not, $\exists a\in\Bbb Z$ such that $f(a) = 1.$

$\Rightarrow$ $2016a^3 - 263a = 1$

$\Rightarrow$ $a(2016a^2-263)$

$\Rightarrow$ ($a = 1$ and $2016a^2 - 263 = 1$)

or ($a = -1$ and $2016a^2 - 263 = -1$)

Both are impossible. End proof.

So is that going under the assumption that you can choose any arbitrary integer k to test for in the range of f? I try doing that for functions I know are onto and yet it doesn't work....so this is why i'm confused.

FuegoJohnson
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  • See whether this helps you: http://math.stackexchange.com/questions/60907/inverse-of-y-x3-x –  Dec 12 '16 at 07:18
  • For continuous functions, the intermediate value theorem will solve it for you if you can show that $f$ hits arbitrarily large positive and negative values. – Patrick Stevens Dec 12 '16 at 07:23
  • I appreciate the input, but that was the kind of stuff I was hoping to avoid. Plus i must prove it using discrete math techniques(that is, proof by contradiction, ect). – FuegoJohnson Dec 12 '16 at 07:25
  • I don't really know what a "discrete math technique" is. Can you give more examples of such techniques please? – Patrick Stevens Dec 12 '16 at 07:31
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    I shall edit in the proof my professor gave for the second equation during our first test, however I have a hard time applying that outside of that problem. – FuegoJohnson Dec 12 '16 at 07:32
  • Proving that something is surjective is very different from proving it is not. To prove $f : X \to Y$ is surjective, you make an argument along the lines of "let $y \in Y$, here's an $x \in X$ such that $f(x) = y$". Proving it's not looks more like "here's a counterexample $y \in Y$, assume there's some $x \in X$ such that $f(x) = y$, show a contradiction". (Of course, this isn't always the way to do it, but it's the "straightforward" one. For more examples, https://www.cs.colorado.edu/~yuvo9296/courses/csci2824/sect19-functions-examples.html – Henry Swanson Dec 12 '16 at 07:57
  • What are the range and domain of $f$? Are you working on the integers? – Patrick Stevens Dec 12 '16 at 07:57
  • Oh my goodness, I can't believe i left out the domain codomain. Very sorry. From integer to integer. I'll edit it back in. @ HenrySwanson you are correct, I did not mean that I am looking for a proof by contradiction, rather just giving an idea of what type of setup my prof wants. My bad. – FuegoJohnson Dec 12 '16 at 07:59

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