Let $f: \mathbb{P}^1 _{\mathbb{C}} \to \mathbb{P}^1 _{\mathbb{C}}$ be a map over $\mathbb{C}$ defined on homogeneous cordinates via
$$[x:y] \to [x^d:y^d]$$
for $d \ge 2$.
My question is why $f$ is a covering of degree $d$?
The main problem is to see why for example the point $[1:0]$ has $d$ preimages in $\mathbb{P}^1$?
Here occurs the problem that for the $d$-th primitive root of unity $\zeta$ all "preimages" $[\zeta^k:0]$ for $k$ integer, $0≤k<d$ are the same point in $\mathbb{P}^1 _{\mathbb{C}}$ by definition / construction of homogeneous coordinates.
Where is an error in my reasoning?