Question

Find the range of the following functions:
$f\left(x\right)=\ln (x-[x\left]\right)$, where $[.]$ denotes the greatest integer function.

Answer 1

Since $x-\left[x\right]=\left\{x\right\}$

And w can say that

Range of $\left\{x\right\}$ is$[0,1)$

So, range of $f\left(x\right)=\ln \left\{x\right\}$ will be

$\left[\ln \right(0^{+}),\ln (1\left)\right]$ or $[\infty ,0]$