Question

Let $f\left(x\right)=\frac{x-\left[x\right]}{1+x-\left[x\right]},xϵR$, where [ x] denotes the greatest integer function. Then, the range of f is

• A. $(0,1)$
• B. $[0,\frac{1}{2})$
• C. $[0,1]$
• D. $[0,\frac{1}{2}]$

Answer 1

The correct option is B $[0,\frac{1}{2})$

The graph of $y=x-\left[x\right]$ is as shown below


When x is an integer, $x-\left[x\right]=0$
Hence, f(x) = 0 when x is an integer
$x\to \left[x\right]$ as x tends to an integer.
Let X = $x-\left[x\right]$
So, $f\left(x\right)=\frac{X}{1+X},Xϵ[0,1)$
As $X\to 1,\frac{X}{1+X}\to \frac{1}{2}$
Hence, the range of f(x) is $[0,\frac{1}{2})$ .