Question

Let $f\left(x\right)=(-1{)}^{\left[x\right]}$ (where [.] denotes the greatest integer function), then

• A. Range of $f$ is $\{-1,1\}$
• B. $f$ is an even function
• C. $f$ is an odd function
• D. $\underset{x\to n}{lim}f\left(x\right)$ exists, for every integer n

Answer 1

The correct option is A Range of $f$ is $\{-1,1\}$

$f\left(x\right)={(-1)}^{\left[x\right]}$
When $0<x<1\Rightarrow \left[x\right]=0$
$f\left(x\right)={(-1)}^0=1$
When $x<0\Rightarrow \left[x\right]=-ve$
If x is even, then $f\left(x\right)=1$
If x is odd, then $f\left(x\right)=-1$
When $x>1\Rightarrow \left[x\right]=+ve$
If x is even, then $f\left(x\right)=1$
If x is odd, then $f\left(x\right)=-1$
$\therefore$ Range of $f\left(x\right)$ is $\{-1,1\}$