Finding the Domain of Functions with Absolute Value and Square Roots

To find the domain of a function, we need to identify all the possible input values (x) for which the function is defined. The given function is:

\({\rm{f}}\left( {\rm{x}} \right) = \frac{1}{{\sqrt {\left| {\rm{x}} \right| - {\rm{x}}} }}\)

For this function to be defined, two conditions must be met:

1. The expression under the square root must be non-negative.
2. The denominator cannot be zero.

Combining these two conditions, the expression under the square root in the denominator must be strictly positive:

\(\left| {\rm{x}} \right| - {\rm{x}} > 0\)

This inequality can be rewritten as:

\(\left| {\rm{x}} \right| > {\rm{x}}\)

We need to find the values of \({\rm{x}}\) that satisfy this inequality. We can analyze this by considering the two cases for the absolute value \({\left| {\rm{x}} \right|}\).

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

If \({\rm{x}} \ge 0\), then \({\left| {\rm{x}} \right|} = {\rm{x}}\). Substituting this into the inequality \({\left| {\rm{x}} \right| > {\rm{x}}}\), we get:

\({\rm{x}} > {\rm{x}}\)

Subtracting \({\rm{x}}\) from both sides gives:

\(0 > 0\)

This statement is false. Therefore, there are no values of \({\rm{x}} \ge 0\) that satisfy the inequality \({\left| {\rm{x}} \right| > {\rm{x}}}\).

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

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

\(-{\rm{x}} > {\rm{x}}\)

Adding \({\rm{x}}\) to both sides gives:

\(0 > 2{\rm{x}}\)

Dividing both sides by 2 (and since 2 is positive, the inequality direction does not change) gives:

\(0 > {\rm{x}}\)

This means \({\rm{x}} < 0\). This result is consistent with our assumption for this case (\({\rm{x}} < 0\)). Therefore, all values of \({\rm{x}} < 0\) satisfy the inequality \({\left| {\rm{x}} \right| > {\rm{x}}}\).

Combining the Cases to Determine the Domain

From Case 1, we found no solutions when \({\rm{x}} \ge 0\). From Case 2, we found that all \({\rm{x}} < 0\) are solutions.

Combining these results, the inequality \({\left| {\rm{x}} \right| > {\rm{x}}}\) is satisfied only when \({\rm{x}} < 0\).

Thus, the domain of the function \({\rm{f}}\left( {\rm{x}} \right) = \frac{1}{{\sqrt {\left| {\rm{x}} \right| - {\rm{x}}} }}\) is the set of all real numbers \({\rm{x}}\) such that \({\rm{x}} < 0\).

In interval notation, this domain is \((-\infty, 0)\).

Let's check a few points:

• If \({\rm{x}} = 1\) (in \([0, \infty)\) case): \({\left| 1 \right| - 1} = 1 - 1 = 0\). \(\sqrt{0}\) is 0, but it's in the denominator, so undefined.
• If \({\rm{x}} = 0\) (in \([0, \infty)\) case): \({\left| 0 \right| - 0} = 0 - 0 = 0\). \(\sqrt{0}\) is 0, in the denominator, undefined.
• If \({\rm{x}} = -1\) (in \((-\infty, 0)\) case): \({\left| -1 \right| - (-1)} = 1 - (-1) = 1 + 1 = 2\). \(\sqrt{2}\) is defined, and the function is \(1/\sqrt{2}\), which is defined.

The domain is indeed \((-\infty, 0)\).

Summary of Domain Analysis

Revision Table: Key Concepts for Function Domain

Additional Information: Understanding Absolute Value Inequalities

Solving inequalities involving absolute values often requires splitting the problem into cases based on the sign of the expression inside the absolute value. For an inequality like \({\left| {\rm{x}} \right| > {\rm{a}}}\), where \({\rm{a}}\) is a constant:

• If \({\rm{a}} \ge 0\), then \({\left| {\rm{x}} \right| > {\rm{a}}}\) is equivalent to \({\rm{x}} > {\rm{a}}\) or \({\rm{x}} < -{\rm{a}}\).
• If \({\rm{a}} < 0\), then \({\left| {\rm{x}} \right| > {\rm{a}}}\) is true for all real numbers \({\rm{x}}\) because \({\left| {\rm{x}} \right|}\) is always non-negative and thus always greater than a negative number.

In our problem, we had \({\left| {\rm{x}} \right| > {\rm{x}}}\). Here, the right side is not a constant, but a variable \({\rm{x}}\). This is why we analyze cases based on the sign of \({\rm{x}}\).

• If \({\rm{x}} \ge 0\), the inequality becomes \({\rm{x}} > {\rm{x}}\), which is false.
• If \({\rm{x}} < 0\), the inequality becomes \(-{\rm{x}} > {\rm{x}}\), which simplifies to \({\rm{x}} < 0\).

Combining these, the solution is \({\rm{x}} < 0\).

This detailed analysis confirms that the domain of the function is indeed \((-\infty, 0)\).