Answer
The value of $y$ in this case is $$y=-\frac{\pi}{4}$$
Work Step by Step
$$y=\tan^{-1} (-1)$$
First, we see that the domain of inverse tangent function is $[-\infty,\infty]$. Therefore, in fact when we deal with inverse tangent function, we do not need to do this checking step.
The range of inverse tangent function is $(-\frac{\pi}{2},\frac{\pi}{2})$. In other words, $y\in(-\frac{\pi}{2},\frac{\pi}{2})$.
We can rewrite $y=\tan^{-1}(-1)$ into $\tan y=-1$
We know that $$\tan\frac{\pi}{4}=1$$ which means $$-\tan\frac{\pi}{4}=-1$$ $$\tan(-\frac{\pi}{4})=-1$$
Also, we see that $-\frac{\pi}{4}\in(-\frac{\pi}{2},\frac{\pi}{2})$
Therefore, the exact value of $y$ here is $$y=-\frac{\pi}{4}$$
