Let $\displaystyle f:S^1\rightarrow X$. Show that $\displaystyle [f]=1 \in \pi_1(X)$ iff $\displaystyle f$ extends to the unit disc $\displaystyle D^2$

Am I right in thinking that the fundamental group of this is trivial, so the identity map is homotopic to any "loop" $\displaystyle f:S^1\rightarrow X$? Any hints are greatly appreciated.