Question

For a function $f\left(x\right)=\frac{2\left(x^2+1\right)}{\left[x\right]}$ (where $\left[.\right]$ denotes the greatest integer function), if $1\le x<4$ , then

• A. the range of $f$ is $\left[4,\frac{17}{2}\right)$
• B. $f$ is bijective function
• C. the maximum value of $f\left(x\right)$ is $\frac{34}{3}$
• D. the minimum value of $f\left(x\right)$ is 4

Answer 1

Answer: (D)

We have, $f(x)=\frac{2\left(x^2+1\right)}{[x]}$
When $x\in [1,2)$, then $f(x)=2\left(x^2+1\right)\Rightarrow R_f=[4,10)$
When $x\in [2,3)$, then $f(x)=x^2+1\Rightarrow R_f=[5,10)$
When $x\in [3,4)$, then $f(x)=\frac{2\left(x^2+1\right)}{3}\Rightarrow R_f=\left[\frac{20}{3},\frac{34}{3}\right)$
$\therefore R_f=\left[4,\frac{34}{3}\right)$
Clearly, for $x\in [1,2)$ and $x\in [2,3)$ range of $f(x)$ has common elements
$\Rightarrow f(x)$ is many-one
$\Rightarrow f(x)$ is not bijective