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)

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