Is there a total binary computable function that specifies Turing machines with nonempty domain?
My Tip for you:
Check out the second recursion theorem and plug $\Psi$ in for $Q$.
If you can't continue on you own here is my solution:
The second recursion theorem implies that there is such a $m$ that:
$$\varphi_m \simeq \lambda n.\Psi(m,n)$$
($\simeq$ means that the left and right expression are equal for every input $n$ equal and if an expression in undefined for a certain $n$ this is also true for the other expression)
Case 1: $\Psi(m,n) = 1$
This implies $f(m) = 0$ and thus $\text{domain}(\phi_m) = \emptyset$.
But notice that $\varphi_m \simeq 1$ is implied by our assumption.
Thus the domain of $\varphi_m$ is $\{1\}$. That's a contradiction.
Case 2: $\Psi(m,n) = \text{undefined}$
This implies $f(m) = 1$ and thus $\text{domain}(\phi_m)\neq\emptyset$.
But once again $\varphi_m \simeq \lambda n.\Psi(m,n) \Rightarrow \varphi_m \simeq \text{undefined} \Rightarrow \text{domain}(\phi_m) = \emptyset$, which is a contradiction.