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Functions — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsFunctions1 Mark
Question
Write the domain and range of function f(x) given by $\text{f(x)}=\frac{1}{\sqrt{\text{x}-|\text{x}|}}$
✓

Answer

We have,
$\text{f(x)}=\frac{1}{\sqrt{\text{x}-|\text{x}|}}$
We know that,
$|\text{x}|=\begin{cases}\text{x},&\text{if x}\geq0\\-\text{x},&\text{if x}<0\end{cases}$
$\Rightarrow\text{x}-|\text{x}|=|\text{x}|=\begin{cases}\text{x}-\text{x}=0,&\text{if x }\geq0\\\text{x}+\text{x}=2\text{x},&\text{if x }<0\end{cases}$
$\Rightarrow\text{x}-|\text{x}|\leq0$ for all x
$\Rightarrow\frac{1}{\sqrt{\text{x}-|\text{x}|}}$ does not take real values for any $\text{x}\in\text{R}$
$\Rightarrow\text{f(x)}$ is not defined for any $\text{x}\in\text{R}$
Hence, domain $(\text{f})=\phi=\text{Range(f)}$

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