Question

Let $f\left(x\right)=\sqrt{\left|x\right|-\left\{x\right\}}$ where $\{\cdot \}$ denotes the fractional part of $x$ and $X,Y$ and its domain and range respectively,then

• A. $f:X\to Y:y=f\left(x\right)$ is one-one function
• B. $X\in (-\infty ,-\frac{1}{2}]\cup [0,\infty )$ and $Y\in [\frac{1}{2},\infty )$
• C. $X\in (-\infty ,-\frac{1}{2}]\cup [0,\infty )$ and $Y\in [0,\infty )$
• D. None of the above

Answer 1

The correct option is C $X\in (-\infty ,-\frac{1}{2}]\cup [0,\infty )$ and $Y\in [0,\infty )$

$f\left(x\right)=\sqrt{\left|x\right|-\left\{x\right\}}$
$\left|x\right|\ge \left\{x\right\}$
$\Rightarrow X\in (-\infty ,-\frac{1}{2})\cup (0,\infty )$
$\Rightarrow Y\in (0,\infty )$ and $f\left(x\right)$ is many-one.