Finding the Domain of a Function with Absolute Value and Square Root

The given function is \({\rm{f}}\left( {\rm{x}} \right) = \frac{1}{{\sqrt {\left| {\rm{x}} \right| - {\rm{x}}} }}\).

To determine the domain of this function, we need to find all real values of \({\rm{x}}\) for which the function is defined. The function involves a square root in the denominator. For the function to be well-defined in the set of real numbers, two conditions must be satisfied simultaneously:

• The expression inside the square root, \({\left| {\rm{x}} \right| - {\rm{x}}}\), must be non-negative (\({\left| {\rm{x}} \right| - {\rm{x}}} \ge 0\)).
• The denominator, \({\sqrt {\left| {\rm{x}} \right| - {\rm{x}}} }\), cannot be zero (\({\sqrt {\left| {\rm{x}} \right| - {\rm{x}}} } \ne 0\), which means \({\left| {\rm{x}} \right| - {\rm{x}}} \ne 0\)).

Combining these two conditions, the expression inside the square root in the denominator must be strictly positive. Thus, we require:

We need to solve this inequality for \({\rm{x}}\). The presence of the absolute value \({\left| {\rm{x}} \right|}\) suggests analyzing the inequality based on the sign of \({\rm{x}}\).

Case 1: \({\rm{x}} \ge 0\)

If \({\rm{x}}\) is greater than or equal to zero, the absolute value of \({\rm{x}}\) is simply \({\rm{x}}\) itself. So, \({\left| {\rm{x}} \right|} = {\rm{x}}\).

Substituting this into the inequality \({\left| {\rm{x}} \right| - {\rm{x}}} > 0\), we get:

This statement \(0 > 0\) is false. This means there are no real numbers \({\rm{x}}\) where \({\rm{x}} \ge 0\) for which the inequality \({\left| {\rm{x}} \right| - {\rm{x}}} > 0\) holds true. Therefore, the function is not defined for any \({\rm{x}} \ge 0\).

Case 2: \({\rm{x}} < 0\)

If \({\rm{x}}\) is less than zero, the absolute value of \({\rm{x}}\) is the negation of \({\rm{x}}\). So, \({\left| {\rm{x}} \right|} = -{\rm{x}}\).

Substituting this into the inequality \({\left| {\rm{x}} \right| - {\rm{x}}} > 0\), we get:

To solve for \({\rm{x}}\), we divide both sides of the inequality by -2. When dividing or multiplying an inequality by a negative number, the direction of the inequality sign must be reversed.

This means the inequality \({\left| {\rm{x}} \right| - {\rm{x}}} > 0\) is true for all real numbers \({\rm{x}}\) that are strictly less than zero.

Determining the Domain of f(x)

From the analysis of both cases, we found that the function \({\rm{f}}\left( {\rm{x}} \right) = \frac{1}{{\sqrt {\left| {\rm{x}} \right| - {\rm{x}}} }}\) is defined only when \({\rm{x}} < 0\).

In interval notation, the set of all real numbers \({\rm{x}}\) such that \({\rm{x}} < 0\) is represented as \((-\infty, 0)\).

Comparing this result with the given options, we see that the domain is \((-\infty, 0)\).

Revision Table: Rules for Finding Function Domains

Here are some general rules to remember when finding the domain of a real-valued function:

• Avoid division by zero: Exclude values of \({\rm{x}}\) that make the denominator of a fraction equal to zero.
• Avoid taking the square root (or any even root) of a negative number: Ensure that the expression under an even root is non-negative.
• Avoid taking the logarithm of a non-positive number: Ensure that the argument of a logarithm is strictly positive.
• Consider piecewise definitions and absolute values: Break down the domain based on the conditions that define different parts of the function or the absolute value expression.

Additional Information: Absolute Value Properties in Inequalities

Understanding the definition of the absolute value function, \({\left| {\rm{x}} \right|}\), is crucial for solving inequalities like the one encountered here. The definition is \({\left| {\rm{x}} \right|} = {\rm{x}}\) if \({\rm{x}} \ge 0\) and \({\left| {\rm{x}} \right|} = -{\rm{x}}\) if \({\rm{x}} < 0\). This piecewise definition requires splitting the problem into cases. For inequalities of the form \({\left| {\rm{x}} \right|} > {\rm{a}}\), the solution is \({\rm{x}} < -{\rm{a}}\) or \({\rm{x}} > {\rm{a}}\) (for \({\rm{a}} > 0\)). For inequalities of the form \({\left| {\rm{x}} \right|} < {\rm{a}}\), the solution is \(-{\rm{a}} < {\rm{x}} < {\rm{a}}\) (for \({\rm{a}} > 0\)). In our specific case, the inequality involves \({\left| {\rm{x}} \right|} - {\rm{x}}\), which requires direct substitution based on the cases \({\rm{x}} \ge 0\) and \({\rm{x}} < 0\).