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Write the Domain and Range of Function F(X) Given by F ( X ) = 1 √ X − | X | . - Mathematics

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Question

Write the domain and range of function f(x) given by

\[f\left( x \right) = \frac{1}{\sqrt{x - \left| x \right|}}\] .
 

Solution

Given:

\[f\left( x \right) = \frac{1}{\sqrt{x - \left| x \right|}}\] We know that \[\left| x \right| = \begin{cases}x, & if x \geq 0 \\ - x, & if x < 0\end{cases}\] \[\Rightarrow x - \left| x \right| = \begin{cases}x - x = 0, & if x \geq 0 \\ x + x = 2x, & if x < 0\end{cases}\]
⇒ x - | x| ≤ 0 for all x.
\[\Rightarrow \frac{1}{\sqrt{x - \left| x \right|}}\] does not take any real values for any x ∈ R.
⇒ f (x) is not defined for any x ∈ R.

Hence,
domain ( f ) = Φ and range ( ) = Φ 

 


 
 
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Chapter 3: Functions - Exercise 3.5 [Page 42]

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RD Sharma Mathematics [English] Class 11
Chapter 3 Functions
Exercise 3.5 | Q 12 | Page 42

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