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 $1\leq x < 4$ , 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