Surjective, injective and bijection problem I have a homework problem that is to find the range that allows the function below to be bijective. $$\ f(x) = x^3 -x^2 -x+1  $$ 
 My problem is I don't know how to go about showing this function is either surjective or injectieve. So far we have worked with simple stuff like $\ x+1 $ or $\ x^2 $ I understand that injective functions are found like this:

$$ \begin{align} 
&amp; f(x) = x^3\\ 
&amp; f(a) = f(b)\\ 
&amp; a^3 = b^3\\ 
&amp; a = b\\
\end{align}$$

and for surjective functions I need  $\ b = f(a) $ so,

$$ \begin{align}
f(a) = b\\
a^3 = b\\
a=\sqrt[3]{b}
\end{align}$$

I am quite confident about finding the domain, but have no idea how to tackle this function.

Thanks for looking.
Since the function is a third degree polynomial, you will have a harder time proving injectivity and surjectivity, but it will be possible. I will first only give you one hint:

Try to plot the function as accurately as possible. Once you have plotted it, you can usually guess if the function is surjective and if it is injective. Once you guess the right answer, it is easier to prove it (since you are not proving the "wrong" answer.

To plot the function, you should find its zeros and the zeros of the derivative, which should not be hard.
A function that is continuous and monotonous increasing or decreasing is injective.
Every polynomial function is continuous everywhere. 

In order to be surjective, just watch out that the codomain of $f$ is equal to its range. As an example, assume that $f(x)$ has domain $D$ and range $R \neq \mathbb{R}$. In that case, $f:D \rightarrow R$ is surjective, but $f: D \rightarrow \mathbb{R}$ is not.