Determine if the function is surjective Let $\\f: \ \mathbb{R} \to \mathbb{R} $ via $ f(x) = \frac {x}{1+x^2}$.

I know this function is not injective, but im not sure how to determine if the function is surjective or not
In order to show that $f$ is surjective, you must show that for any $y\in\Bbb R,$ there is some $x\in\Bbb R$ such that $f(x)=y.$ That is, we must be able to solve $$\frac{x}{1+x^2}=y$$ for $x$ and find that $x\in\Bbb R.$ The quadratic formula will come in handy in your attempts to do so.
A function is surjective if every element has a pre-image. I will show that this function is not surjective by showing $2$ does not have a pre-image.

Indeed, if $2$ has a pre-image, say $x$ then $$\frac{x}{1+x^2}=2$$
Then $x=2+2x^2$. That is $$2x^2-x+2=0$$
As $1-4(2)(2)&lt;0$. Therefore, this equation has no real roots. 
Hence, $f$ cannot be surjective.