Find the Domain and Range of the Function
Question:
Write the domain and range of the function
\[ f(x)=\frac1{\sqrt{x-|x|}} \]
Solution:
Since square root is in denominator,
\[ x-|x|>0 \]
Case I: \(x\ge0\)
\[ |x|=x \]
\[ x-|x|=0 \]
Not allowed.
Case II: \(x<0\)
\[ |x|=-x \]
\[ x-|x|=x+x=2x<0 \]
Not allowed.
Therefore, no real value satisfies the condition.
Domain:
\[ \boxed{\phi} \]
Since domain is empty, range is also empty.
Range:
\[ \boxed{\phi} \]