Question:
Let f : $\mathbb{R}$ $\to$ $\mathbb{R}$ via $ f(x) = \frac{x}{1+x^2}$. Is $f$ injective?
My attempt:
I am unable to find a counter example to prove that it is not injective.
Suppose $ f(a) = f(b)$ for some $a,b \in \mathbb{R}$.
$ \frac{a}{1+a^2}=\frac{b}{1+b^2}$
$a\ +\ ab^2\ =\ b\ +ba^2$
$ ab^2-b\ \ =ba^2-a$
$ b\left(ab-1\right)=a\left(ab-1\right)$
$ \left(b-a\right)\left(ab-1\right)=0$
Would this imply that $ a = b$? In questions where I cannot easily figure out a counter example to prove a function is not injective, what should I do? Should I try to prove that it is injective and then reach a contradiction?