Key Concepts and Formulas

• Domain of a Function: The set of all possible real input values ($x$) for which the function is defined and yields a real output.
• Restrictions for Square Roots: The expression under a square root must be non-negative ($\ge 0$).
• Restrictions for Fractions: The denominator of a fraction cannot be zero ($\ne 0$).
• Definition of Absolute Value:
  • $|x| = x$ if $x \ge 0$
  • $|x| = -x$ if $x < 0$

Step-by-Step Solution

Step 1: Identify the restrictions on the function. The given function is $f(x) = \frac{1}{\sqrt{|x| - x}}$. For this function to be defined, two conditions must be met:

1. The expression inside the square root must be non-negative: $|x| - x \ge 0$.
2. The denominator cannot be zero: $\sqrt{|x| - x} \ne 0$.

Combining these two conditions, the expression inside the square root must be strictly positive: $|x| - x > 0$

Step 2: Solve the inequality $|x| - x > 0$. We can rewrite the inequality as: $|x| > x$ To solve this inequality, we consider two cases based on the definition of the absolute value of $x$.

Step 3: Analyze Case 1: $x \ge 0$. If $x \ge 0$, then by the definition of absolute value, $|x| = x$. Substituting this into the inequality $|x| > x$: $x > x$ Subtracting $x$ from both sides gives: $0 > 0$ This statement is false. Therefore, there are no values of $x$ in the interval $[0, \infty)$ that satisfy the inequality.

Step 4: Analyze Case 2: $x < 0$. If $x < 0$, then by the definition of absolute value, $|x| = -x$. Substituting this into the inequality $|x| > x$: $-x > x$ Add $x$ to both sides of the inequality: $0 > 2x$ Divide both sides by 2. Since 2 is positive, the inequality sign remains the same: $0 > x$ This can be written as $x < 0$. This result is consistent with our assumption for this case ($x < 0$). Therefore, all values of $x$ such that $x < 0$ satisfy the inequality.

Step 5: Combine the results from both cases. From Case 1 ($x \ge 0$), we found no solutions. From Case 2 ($x < 0$), we found that all $x < 0$ are solutions. Thus, the domain of the function $f(x)$ is the set of all real numbers $x$ such that $x < 0$.

Step 6: Express the domain in interval notation. The condition $x < 0$ corresponds to the interval $(-\infty, 0)$.

Common Mistakes & Tips

• Forgetting the strict inequality: When an expression is in the denominator of a square root, the expression must be strictly greater than zero (not just non-negative) because $\sqrt{0} = 0$, which would make the denominator zero and the function undefined.
• Incorrectly applying absolute value definition: Ensure you use the correct definition of $|x|$ for different cases of $x$. For example, if $x$ is negative, $|x|$ is $-x$, which is a positive value.
• Algebraic errors in inequalities: Be careful when manipulating inequalities, especially when multiplying or dividing by negative numbers, as this reverses the inequality sign.

Summary

To determine the domain of $f(x) = \frac{1}{\sqrt{|x| - x}}$, we established that the expression under the square root in the denominator must be strictly positive, i.e., $|x| - x > 0$, which simplifies to $|x| > x$. By considering the cases where $x \ge 0$ and $x < 0$, we found that the inequality $|x| > x$ is only satisfied when $x < 0$. Therefore, the domain of the function is $(-\infty, 0)$.

The final answer is $\boxed{\left( { - \infty ,0} \right)}$.