Question

If $f:[0,2]\to A,f\left(x\right)=\left[x^2\right]-[x{]}^2$ is a real valued function, then minimum elements required in set $A$ is
(where [.] denotes greatest integer function)

• A. $(-\infty ,0]$
• B. $(-\infty ,1]$
• C. $\{0,1,2\}$
• D. $N$

Answer 1

The correct option is C $\{0,1,2\}$

$f\left(x\right)=\left[x^2\right]-[x{]}^2$
We know that,
$x\in [0,2]\Rightarrow x^2\in [0,4]$
We have to check for integer values of $x^2$ in $[0,4]$
Now,
$f\left(0\right)=0f\left(1\right)=1-1=0$
$f\left(\sqrt{2}\right)=2-1=1f\left(\sqrt{3}\right)=3-1=2$
$f\left(2\right)=4-4=0$
So, $f\left(x\right)$ is function when
$A=\{0,1,2\}$