Restriction of a function and surjectivity

I have a function $f:[-1,1] \rightarrow \mathbb R$. If $f$ is surjective then restriction $f_{|[0,1]}:[0,1] \rightarrow \mathbb R $ is surjective?

if I consider the function $f(x)=\tan({{\pi x} \over {2} })$ is surjective in $(-1,1)$ but$f_{|[0,1]}:[0,1) \rightarrow \mathbb R $ is not surjective so the question is false in general? or exist functions surjective with restriction surjective?