Finding the Domain of a Function

The domain of a function is the set of all possible input values (often represented by \(x\)) for which the function is defined and produces a real output.

For the given function \(f\left( x \right) = \frac{1}{{\sqrt {\left| x \right| - x} }}\), we need to consider two main conditions for the function to be defined:

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. That is, we must have:

\(\left| x \right| - x > 0\)

To solve this inequality involving the absolute value, we consider two cases based on the definition of \(|x|\).

Case 1: \(x \ge 0\)

If \(x \ge 0\), then \(|x| = x\). Substitute this into the inequality:

\(x - x > 0\)

\(0 > 0\)

This statement is false. Therefore, there are no values of \(x\) in the interval \([0, \infty)\) for which the function is defined.

Case 2: \(x < 0\)

If \(x < 0\), then \(|x| = -x\). Substitute this into the inequality:

\(-x - x > 0\)

\(-2x > 0\)

To solve for \(x\), divide both sides by -2. Remember to reverse the inequality sign when dividing by a negative number:

\(\frac{-2x}{-2} < \frac{0}{-2}\)

\(x < 0\)

This inequality \(x < 0\) is consistent with the condition for this case (\(x < 0\)). Therefore, all values of \(x\) such that \(x < 0\) satisfy the original inequality \(|x| - x > 0\).

Combining the Cases

From Case 1, there are no solutions for \(x \ge 0\). From Case 2, the solutions are \(x < 0\).

Combining these results, the function \(f(x)\) is defined only when \(x < 0\).

In interval notation, \(x < 0\) is represented as \((-\infty, 0)\).

Comparing with Options

Let's compare our result with the given options:

Our calculated domain, \((-\infty, 0)\), matches Option 1.

Conclusion

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

Revision Table: Understanding Domain Constraints

Additional Information: Absolute Value and Inequalities

The absolute value of a number \(x\), denoted by \(|x|\), is its distance from zero on the number line. It is defined as:

\(|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}\)

When solving inequalities involving absolute values, it is often necessary to consider the cases where the expression inside the absolute value is non-negative and where it is negative, as we did in the solution. This case-by-case analysis helps correctly apply the definition of the absolute value.

For example, solving \(|x| > a\) (where \(a > 0\)) is equivalent to \(x > a\) or \(x < -a\). Solving \(|x| < a\) (where \(a > 0\)) is equivalent to \(-a < x < a\).

In our problem, the inequality \(\left| x \right| - x > 0\) is not a standard form like \(|x| > a\) or \(|x| < a\), hence the case analysis based on the sign of \(x\) is the appropriate method.