Question

Let $f:(4,6)\to (6,8)$ be defined by $f\left(x\right)=x+\left[\frac{x}{2}\right]$, where $[.]$ represents the greatest integer function. Then the range of $f$ is

• A. $[7,8)$
• B. $(7,8)$
• C. $(6,8)$
• D. $(6,7]$

Answer 1

The correct option is C $(6,8)$

$f\left(x\right)=x+\left[\frac{x}{2}\right]$
Now, $4<x<6\cdots \left(1\right)$
$\Rightarrow 2<\frac{x}{2}<3\Rightarrow \left[\frac{x}{2}\right]=2\cdots \left(2\right)$

Adding $\left(1\right)$ and $\left(2\right)$, we get
$6<x+\left[\frac{x}{2}\right]<8\Rightarrow 6<f\left(x\right)<8$
$\therefore$ Range of $f$ is $(6,8)$