Q:

Domain of $( \frac{x-1}{x-2\{x\}})^{1/2}$

Find the Domain of $\left(\frac{x-1}{x-2\{x\}}\right)^{1/2}$ where $\{\}$ denotes the fractional part function.
$\left(\frac{x-1}{x-2\{x\}}\right)$ should be positive,
$\left(\frac{x-1}{x-2\{x\}}\right)\ge0$.
The denominator can't be zero, so $x≠0$. I can't do anything else.
How can I proceed$?$

A:

Note that $\{ x \} $ will always be less than 1. Therefore, $2\{ x \} $  will be less than 2. Thus, $x-2\{ x \} $ will be positive for $x>2$.
We then check 2 itself. Replacing $x$ with 2, it is clear that the function is defined as numerator and denominator are positive. 
Next, we can look at the interval $(1, 2)$. On that interval, the $\{ x\} $ is just $x-1$, making the denominator $2-x$, which is always positive on $(1,2)$. Since the numerator is positive here also, the function is defined. 
At 1 the function is defined. 
Next, check $(0,1)$. The denominator is just $x-2x=-x$. So the denominator is negative, as is the numerator. Thus the function is defined.
Continue like this into negative intervals and see if you can finish a similar argument! 

