Let $f:(S^1,1)\to(S^1,1)$ a continuous function. Then calculate: $f_{*}(\pi_1(S^1,1))$ and $\pi_1(f(S^1),1)$, where $S^1=\{z\in\mathbb{C}:|z|=1\}$ and if $[\alpha]\in\pi(X,x)$ then $f_{*}([\alpha])=[f\circ\alpha]\in\pi_1(X,x)$.
I know that $S^1=\langle[\omega_1]\rangle$ where $\omega_1:[0,1]\to S^1:s\mapsto e^{2\pi is}$ but I don't know how to use this ideas.