Question: I am doing homework for an Intro to Proofs class, so would appreciate general information or hints only rather than a specific solution
The question is:
Determine which of the following functions $$f:\mathbb{R}\to\mathbb{R}$$ are injective, which are surjective and which are bijective. 

The specific function I'm having trouble with is:
$$f\left(x\right)=x^{3}-x$$

I can easily show that $f$ is not injective, by noting that $f\left(-1\right)=f\left(0\right)=f\left(1\right)=0$

Up until now, I would let $y\in\mathbb{R}$ , the codomain of $f$, and then manipulate the equation $$y=x^3-x$$ to get something of the form $x=\dots$, to show that there exists an element $x$ in the domain such that $$f\left(x\right)=y$$

I can factorize the equation as follows:
$$y=x\cdot\left(x+1\right)\cdot\left(x-1\right)$$

But I still can't tell where to go from here. I can tell from a graph of the function that $f$ is surjective, but can't figure out how to prove it.

Answer: 
As an alternate approach, if $y\in \mathbb{R}$, then the fundamental theorem of algebra (and the conjugate root theorem) tells us that $g(x)=x^3-x-y$ has at least one real root (because complex roots come in conjugate pairs). Thus, there is an $x\in \mathbb{R}$ such that \begin{align}x^3-x-y&=0\\ 
\implies x^3-x&=y\\
\implies f(x)&=y\end{align}
Two hints:

Hint 1: Show that for M > 0, there exists an $x\_0$ such that

$f(-x\_0) < -M < M < f(x\_0)$

Hint 2: I found this [history of math on Simon Stevin](https://en.wikipedia.org/wiki/Intermediate_value_theorem#History) interesting.

So, if you know nothing about continuous functions, you are OK - you can also use the Intermediate Value Theorem for Polynomials (or derive it for your classwork).