# Domain of a function with fractional part expression

It goes like : find domain of $$f(x)= \frac{1}{\sqrt{(\{x+1\}-x^{2}+2x)}}$$ where {.} denotes fractional part of x. I went on with the usual methodology putting denominator greater than 0,

$$\{x+1\}-(x)^{2}+2x>0$$

I changed fractional part to (x+1)-[x+1] where [.] denoted greatest integer function.

solving further,
$$3x-(x)^{2}+1>[x+1]$$
$$3x-(x)^{2}>[x]$$
after this i tried making graph of this but to no avail as i couldn't specify the exact points satisfying the condition. Algebraically I'm not able to make out the next step. Any clues will be helpful,thanks in advance!

• This can help, perhaps: For one thing you need to have $(x^2 - 2x<1)$ to make sure that the denominator is not negative (otherwise it is negative and the square root is not defined). Then you can break into cases, try considering one case: $x^2-2x<-1$ and then separately another case: $-1<x^2-2x<1$
– them
Aug 20 '19 at 9:56

Hint:

Recall that $$x-1<[x]$$ for all real $$x.$$