Browse Questions

# Range of $f(x)=\large\frac{x-[x]}{1+x-[x]}$ where $[x]$ is greatest integer function.

(A) $[0,1]$

(B) $[0,\large\frac{1}{2}$$] (C) [0,\large\frac{1}{2}$$)$

(D) $(0,1)$

$f(x)=0$ if $x=[x]$ that is when $x\in Z$
When $x$ is not an integer then
$0<2(x-[x])<1+x-[x]$
$\Rightarrow\:0<\large\frac{x-[x]}{1+x-[x]}<\frac{1}{2}$
$i.e., 0<f(x)<\large\frac{1}{2}$
$\Rightarrow\:Range = [0,\large\frac{1}{2})$