The Kernel of a Homomorphism

The point is that $f(a)=f(b)$ if and only if $ab^{-1}\in \ker f$. So, if you have checked beforehand that $\ker f=\{e\}$, then $f(a)=f(b)$ automatically implies that $a=b$, which means that $f$ is injective. And it works backwards too, if $f$ is injective, then $\ker f=\{e\}$, because $f(a)=e =f(e)$ implies $a=e$.

If $c$ belongs to the image, then there is a $g_0\in G$ such that $f(g_0)=c$. But then$$\{g\in G\mid f(g)=c\}=\{g_0h\mid h\in\ker f\}.$$