Can anyone please let me know if I did something wrong here?
a) $f(z)=\dfrac{e^2+e^{-z}} {z^2+\bar{z}^2}$.
In this case, we must have $z^2+\bar{z}^2\neq0$, that means $\text{Re}z \neq0$.
b) $f(z)=\dfrac{z^2+5z} {e^z-1}$.
$e^z-1\neq0 \Rightarrow e^z\neq1\Rightarrow e^z\neq e^0\Rightarrow z\neq 0+2k\pi i$. Then, $z\neq 2k\pi i$.
c) $f(z)=\text{Log}(e^z-e^{-z})$.
By definition, $e^z-e^{-z}\neq0 \Rightarrow e^z \neq e^{-z}$ and then, $z\neq - z+2k\pi i \Rightarrow 2z\neq 2k\pi i\Rightarrow z\neq k\pi i$.
Thanks for your attention.
a) $z^2+\bar{z}^2\neq0$ means that $\operatorname{Re}z^2 \neq0$.
b) and c) are correct.