Use limits and calculus to show that $f$ is a bijection I have the following exercise for discrete mathematics:
Show that $f(x)=x^3$ (real-valued) is a bijection. So I have to show that the function is both surjective and injective. So, I know how to do this but I was thinking about an alternative way to show these properties. Are they alright?
Injective
$f'(x)=3x^2$ whenever $x > 0$, $ \text{  } f'(x) > 0$ so it is increasing or decreasing. Whenever $x < 0, \text{ }$ $f'(x) > 0$ so its either increasing or decreasing. Only when $x = 0, f(x) = 0$. So it is impossible that a value in the image is mapped to more than once. 
Surjective
$\lim_{x \rightarrow \infty} f(x) = \infty$ and $\lim_{x \rightarrow -\infty} f(x) = -\infty$ so it must be surjective, since it will reach all values in the codomain.
 A: Taking @Yanko 's comments into consideration, I am editing.
Edit: (for surjectivity)
WTS $\forall a \in \mathbb{R}$ (range), $a$ is in the image of $f$.
Because $f$ is continuous and necessarily non-decreasing (as discussed in a possible proof for injective) then there must exist some $k_1, k_2 \in \mathbb{R}$ s.t. $k_1 < a$ and $a < k_2$. It follows from the intermediate value theorem that $f$ takes the value $a$ for some $x_0$ in the domain. 
Previously:
Hint (for surjectivity): 
Let $a \in \mathbb{R}$. Can we solve $a = x^3$ for $x$? What does this tell you about the mapping $f$?
A: Your injectivity proof looks good (except for the typo on the derivative), but you need to say something about $f$ being continuous on $\mathbb{R}$ if you want to use the fact that the limits go to $\pm\infty$ in order to conclude that $f$ is surjective. 
A: Your answers are good but here are some things I would add/correct:
Injective:
The derivative of $f$ is not $2x^3$, it is $3x^2$. So $f'(x)>0$ for all $x\not = 0$ and so it is always increasing. Note that a function can increase then decrease while not being injective. (for example $f(x)=x^2$ decrease then increase but is not injective because $f(-1)=f(1)$).
Surjective:
You need to mention that you're using the Intermediate Theorem.
