Domain and range of a function defined by $\mathbb{C} \rightarrow \mathbb{R}$

Question

Find the domain and range for a function $f$ defined by $f(z)=\log(h(z))$ where $\lbrace h: \mathbb{C} \rightarrow \mathbb{R} \: \vert \: h(z) = \operatorname{Re}(z) + \operatorname{Im}(z) \rbrace$

Here's what I've done:

Simplifying the above expression, we have $f(x,iy) = \log(x+y)$ where $x,y\in\mathbb{R}$.

Thus $x+y>0 \Rightarrow \operatorname{Re}(z)+\operatorname{Im}(z)>0$

$\operatorname{Im}(z)>-\operatorname{Re}(z)$

$\therefore y=-x \Rightarrow z = x-xi = x(1-i) \\ $

So the domain is defined by the set:

$A = \lbrace x(1-i) \: \vert \: x \in \mathbb{R} \rbrace$

If I've done this correctly and the domain is set $A$, then is the range $\mathbb{R}$? If not, how would I go about finding that?

Thanks

Answer 1

The domain of $f$ is $$\{z \in \mathbb{C} \ \big| \ \operatorname{Im}(z) + \operatorname{Re}(z) > 0\}$$

It corresponds to the upper right open half of the complex plane (seen as $\mathbb{R}^2$), split in two parts by the line of equation $y = -x$. This line is the set $A = \lbrace x(1-i) \: \vert \: x \in \mathbb{R} \rbrace$ that you mention.