when does Lie group homomorphism preserve the Haar measure?

Question

Let $G$ be a connected Lie group and $f:G\rightarrow G$ be a Lie group homomorphism such that $df_{e}$, the derivative of $f$ at the identity element $e$, satisfies $|\rm{det}\;df_{e}|=1$. If $\mu$ is a Haar measure on $G$, is it necessarily true that $f$ preserves the measure $\mu$?