Answer
The value of $y$ in need to find 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 $\tan y=1$
In the range $(-\frac{\pi}{2},\frac{\pi}{2})$, we find that only $\tan\frac{\pi}{4}=1$
That means, the exact value of $y$ here is $$y=\frac{\pi}{4}$$
