Question

lf $f:R\to R$ is defined by $f\left(x\right)=\left[2x\right]-2\left[x\right]$
for $x\in R$ where $\left[x\right]$ is the greatest integer not exceeding $x$, then the range of $f$ is

• A. $\{x\in R:0\le x\le 1\}$
• B. $\{0,1\}$
• C. $\{x\in R:x>0\}$
• D. $\{x\in R:x\ge 0\}$

Answer 1

The correct option is B $\{0,1\}$

for all $xϵz;f\left(x\right)=0$
If x is not an integer , then $x=\left[x\right]+(0,1)$
$\therefore f\left(x\right)\left[2\right[x]+2(0,1\left)\right]-2\left[x\right]$
$=\left[2\right(0,1\left)\right]=\left[\right(0,2\left)\right]$
which can take values 0,1
$\therefore$ range of f is $\{0,1\}$