Domain of a function $f(x)= \sqrt{| x |} \cdot(2-\ln x^2)$

Question

Is the domain of the function $f(x)= \sqrt{|x|}\cdot(2-\ln x^2)$ the set $\mathbb{R}\setminus\{0\}$ or $(0, \infty)$ ?

Answer 1

The domain of $\sqrt{f(x)}$ is $\{x \in \mathcal{D}(f) \mid f(x) \geq 0\}$, where $\mathcal{D}(f)$ is the domain of $f$.

The domain of $\ln g(x)$ is $\{x \in \mathcal{D}(g) \mid g(x) > 0\}$.

Furthermore, the domain of a product of functions is the intersection of the domains of the individual functions. That is $\mathcal{D}(pq) = \mathcal{D}(p)\cap\mathcal{D}(q)$.

So, what do you think?

Answer 2

Since there are no problems when $x<0$, (both $\ln{x^2}$ and $\sqrt{|x|}$ are well defined functions), it would be the former.