Let $A$ be a $C^*$-algebra. Denote by $SA$ the suspension of $A$, $SA:=C_0((0,1),A)$.
If we define $F: SA \to SA$ by $(F(f))(t)=f(1-t)$ then $F$ induces group homomorphisms $$K_0(F):K_0(SA)\to K_0(SA) \ \ \ \text{ and }K_1(F):K_1(SA)\to K_1(SA).$$
Show that $K_0(F)=-id_{K_0(SA)}$ and similarly $K_1(F)=-id_{K_1(SA)}$
I've tried to figure out which $^*$-homomorphisms from $SA$ to itself induce the homomorphism $-id$ on the $K_i$-groups. Then just finding a homotopy between $F$ and these homomorphisms will be suffices, but I didn't succeed to find them.
Any help would be appreciated.