$$f(x)= rx-\frac{x}{1+x}$$
Find the values of $r$ at which bifurcation occus and classify those as saddle node, transcritical, or pitchfork bifurcation.
I found the fixed points as $x^*=0,\frac{1}{r}-1$.
I am not sure how to identify what TYPE of bifurcation it is out of the three types. Is there a way to test which one?