Question

Let $f:[2,3]\to B$ be a function defined by $f\left(x\right)=\left[{\log }_2\right[x^2+2x-1\left]\right]$, where $[.]$ represents the greatest integer function. If range of $f$ is $B$, then

• A. $1\in B$
• B. $2\in B$
• C. $3\in B$
• D. $4\in B$

Answer 1

Answer: (B), (C)

The correct options are
B $2\in B$
C $3\in B$

$f\left(x\right)=\left[{\log }_2\right[x^2+2x-1\left]\right]\Rightarrow f\left(x\right)=\left[{\log }_2\left(\right[(x+1{)}^2]-2)\right](\because [x+m]=[x]+m\forall m\in Z)$

If $x\in [2,3]$,
then $(x+1{)}^2\in [9,16]$
For $(x+1{)}^2\in [9,10)$,
$f\left(x\right)=\left[{\log }_{2}7\right]=2$

For $(x+1{)}^2\in [10,16]$,
$f\left(x\right)=3$

$\therefore B=\{2,3\}$