Find the maximal domain and range of:
$f(x) = \ln(x^2 - 7)$
I reasoned that the domain can be found by finding the values of $x$ such that $x^2 - 7 > 0$ so that the logarithm is defined. So:
$x^2 > 7\\ x > \pm \sqrt{7}$
Shouldn't I have arrived at $x > \sqrt{7}$ and $x < -\sqrt{7}$ somehow? From inspection I can see this, but I couldn't seem to understand why I didn't arrive at that algebraically.
Also, by inspection, I would guess that the range is $\{y \in \mathbb{R}\mid y > 0\}$. Is it standard to just derive this from a graph? or should I discuss the limiting behaviour as $x \rightarrow \pm~ \infty$ and as $x \rightarrow \pm\sqrt{7}$ from $+$ and $-$ respectively?