$f:\mathbb{R}^2 -\{0\} \to \mathbb{R}^2 -\{0\} $ such that $f(x,y)=(x^2, y)$
I believe this isn't nulhomotopic since I can't pass through 0, but I don't know how to justify this. I think I should solve this using that composition of a nulhomotopic function with an arbitrary function is nulhomotopic, but still don't know how to apply this.
Any help would be very appreciated. Thank you!