If a map from an n-sphere to itself has no fixed points, it's degree is $(-1)^{n+1}$

Question

This is an exercise problem in Bredons book "Topology and Geometry"

If $f:S^{n} \rightarrow S^{n}$ has no fixed points, $(-f)(x)\neq -x$ for all $x$. Thus $(-f)\simeq -1:S^{n}\rightarrow S^{n}:x\rightarrow-x$.

This implies that their degrees are equal:

$deg((-1)\circ f) = deg(-1)=deg(-1)*deg(f)$

since $deg(-1) \neq 0$ we conclude that $deg(f) = 1$.

Where did I make a mistake?