Find the Domain of \(f(x)=\dfrac{1}{\sqrt{|x|-x}}\)

Solution

Given

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

Since the square root appears in the denominator, the quantity inside the root must be strictly positive.

\[ |x|-x>0 \]

Case 1: \(x \ge 0\)

For \(x \ge 0\),

\[ |x|=x \]

Therefore,

\[ |x|-x=x-x=0 \]

But denominator cannot be zero.

Hence, no value of \(x \ge 0\) belongs to the domain.

Case 2: \(x < 0\)

For \(x < 0\),

\[ |x|=-x \]

Therefore,

\[ |x|-x=-x-x=-2x \]

Since \(x<0\),

\[ -2x>0 \]

Hence all negative real numbers satisfy the condition.