Surjective function proof

How do I prove that this is surjective? I already proved that it's injective.

Also, if a function in both injective and surjective, is that function automatically invertible?

Let $\displaystyle f : [0, +\infty) \rightarrow (0, 1]$ be the function from the set $\displaystyle [0, +\infty)$ to the set

(0, 1] defined such that

$\displaystyle f(x) =\dfrac{1}{1 + x^2}$

for all $\displaystyle x \in [0, +\infty)$, where

$\displaystyle [0, +\infty) = \{x \in \mathbb{R} : 0 \leq x < +\infty\}$,$\displaystyle (0, 1] = \{x \in \mathbb{R} : 0 < x \leq 1\}$