Problem: Let
$$f(x)= \dfrac {1}{\sqrt{|\bigl[|x|-1\bigr]|-5}}$$
where $\bigl[.\bigr]$ is greatest integer function. Find domain of $f(x)$.
Solution: The function $f$ is defined for $|\bigl[|x|-1\bigr]|-5>0$. So
$$|\bigl[|x|-1\bigr]|>5$$
$$5>\bigl[|x|-1\bigr]>-5$$
I don't know whether I am doing right or wrong
Best Answer
You have it all correct until the very end! You're right to restrict $|[|x|-1]| - 5 > 0$. However, at the end, you conclude that $5 > [|x|-1] > -5$, when in fact $|[|x|-1]| > 5$ implies that $[|x|-1] > 5$ or $[|x|-1] < -5$.
From here, we note that $[|x| - 1] = [|x|] - 1$, so we want to see where $[|x|] > 6$ and where $[|x|] < -4$. Since $|x| \geq 0$, the latter is clearly impossible. We thus see that we must have $[|x|] > 6$, which means that $x \geq 7$ or $x \leq -7$ (since we have $[|x|]$ is strictly greater than 6.)