Finding kernel in a mapping

Question

Let $f:\mathbb{C} \rightarrow \mathbb{R}$ where $f(a+bi)=b$

I have already shown that it is indeed a homomorphism.

To find the kernel I believe that it is the set of elements in $\mathbb{C}$ that are mapped onto the identity element in $\mathbb{R}$ by the homomorphism of $f$

But i'm confused about what elements can do this in this specific mapping?

Answer 1

$a+ib \in ker (f) \iff f(a+ib)=0 \iff b=0 \iff a+ib=a \iff a+ib \in \mathbb R$.

Henc $ker(f)= \mathbb R$.