Question

Let $f:(4,6)\to (6,8)$ be a function defined by $f\left(x\right)=x+\left[\frac{x}{2}\right]$ (where [.] denotes the greatest integer function),
then $f^{-1}\left(x\right)$ is equal to

• A. $X-\left[\frac{x}{2}\right]$
• B. $-x-2$
• C. $x-2$
• D. $\frac{1}{x+\left[\frac{\pi }{2}\right]}$

Answer 1

The correct option is C $x-2$

Since, $xϵ(4,6)$
$\therefore f\left(x\right)=x+\left[\frac{x}{2}\right]$
$\Rightarrow f\left(x\right)=x+2,xϵ(4,6)$
$\therefore$ for inverse of $f\left(x\right)$, Let $f\left(x\right)=y$
i.e, $y=x+2\Rightarrow x=y-2$
or $f^{-1}\left(y\right)=y-2$
$\Rightarrow f^{-1}\left(x\right)=x-2$