I'm reading Differential Topology by Guillemin & Pollack, and I would like to know how to prove these facts about fixed points on the sphere in this viewpoint:)

Let $f,g \in {C^0}({\mathbb{S}^{k}},{\mathbb{S}^{k}})$.

(1) If $\deg f \ne {( - 1)^{k + 1}}$, then there exists $x\_0$ s.t. $f(x\_0)=x\_0$.

(2) If $\deg f \ne 1$, then there exists $x\_0$ s.t. $f(x\_0)=-x\_0$.

(3) if $k$ is even, then one of the three maps $f,g,g \circ f$ has a fixed point.

I've been working on them for a while but still can't see the path. Maybe my understanding is not deep enough... Could anyone give me some instructions? Any help will be appreciated:)