Define a homomorphism as $f: G \rightarrow H$ where $f(x)=e^{2\pi i x}$
$G=\mathbb R$ and $H=\mathbb C\setminus\{0\}$
Checking that it is a homomorphism:
$f(xy)=e^{2\pi i x y}=(e^{2πix})^y=f(x)^y$ this is therefore not a homomorphism.
Is my method suitable here?
The group $G$ is here $(\Bbb R,+)$, so that you need to verify $$ f(x+y)=f(x)f(y). $$ But this is true. Note that $H=(\Bbb C^{\times},\cdot)$.