Question

If $f\left(x\right)=\frac{x^{2} - \left[x^{2}\right]}{1 + x^{2} - \left[x^{2}\right]}$ (where $\left[.\right]$ represents the greatest integer part of $x$ ), then the range of $f\left(x\right)$ is

• A. [0,1)
• B. $\left(- 1,1\right)$
• C. $\left(0 , \in fty\right)$
• D. $\left[0, \frac{1}{2}\right)$

Answer 1

Answer: (D)

$x^{2}-\left[x^{2}\right]=\left\{x^{2}\right\}$
$\therefore f\left(x\right)=\frac{\left\{x^{2}\right\}}{1 + \left\{x^{2}\right\}}=\frac{1 + \left\{x^{2}\right\} - 1}{1 + \left\{x^{2}\right\}}=1-\frac{1}{1 + \left\{x^{2}\right\}}$
$\because 0\leq \left\{x^{2}\right\} < 1\Rightarrow 1\leq \left\{x^{2}\right\}+1 < 2$
$\frac{1}{2} < \frac{1}{\left\{x^{2}\right\} + 1}\leq 1\Rightarrow -\frac{1}{2}>\frac{- 1}{1 + \left\{x^{2}\right\}}\geq -1$
$\frac{1}{2}>1+\frac{\left(- 1\right)}{1 + \left\{x^{2}\right\}}\geq 0\Rightarrow $ Range of $f\left(x\right)\in \left[0 , \frac{1}{2}\right]$