Question

The function $f:R\to R$ defined by $f\left(x\right)=[x{]}^2+[x+1]-3,$ (where [.] represents the greatest integer function). Then which of the following is/are true?$

• A. $f\left(x\right)$ one-one
• B. $f\left(x\right)$ many-one
• C. $f\left(0.5\right)=-2$
• D. $f\left(0.7\right)=0$

Answer 1

Answer: (B), (C)

$f\left(0.5\right)=-2$

$f\left(x\right)=[x{]}^2+[x+1]-3$
$f\left(x\right)=[x{]}^2+[x]+1-3$
$f\left(x\right)=\left(\right[x]-1)\left(\right[x]+2)$
Clearly $f\left(0.1\right)=f\left(0.2\right)=f\left(0.5\right)=-2$
So. $f$ is many-one.
$f\left(0.7\right)=(0-1)(0+2)=-2$
Clearly, $f\left(0.7\right)\ne 0$