I am asked to show in a question for a specific map $f: \mathbb{R} \rightarrow \mathbb{C}$ that:
$f(1_\mathbb{R}) = 1_\mathbb{C}$
I understand that $1_\mathbb{R}$ means the identity function $f: X \rightarrow X$, but how do you take a function of an identity function and show it's an identity fuction? Can you provide an example on how to do this?
The specific example given is to show for $f: \mathbb{R} \rightarrow \mathbb{C}$ given by $f(x) = (x,0)$ to show that $f(1_\mathbb{R}) = 1_\mathbb{C}$. Can I get some insight on how to do this?
Your title indicates that you want to show that your homomorphism takes the identity to the identity. All homomorphisms do this.
Thus just go ahead and show you have homomorphism, as one approach.
More succinctly, just use the function's definition. Clearly $1_{\Bbb R}\mapsto 1_{\Bbb C}$.