Finding the Domain of f(x) = 1/√(|x| - x)

The domain of a function is the set of all real numbers x for which the function is defined. For this function, we have two conditions to satisfy:

1. The expression under the square root, |x| - x, must be greater than 0 (since it's in the denominator and we cannot take the square root of a negative number).
2. The entire denominator, √(|x| - x), cannot be equal to 0.

Let's analyze the expression |x| - x based on the value of x.

Step 1: Analyze the Expression |x| - x

The absolute value function |x| is defined as:

$|x|=\left\{\begin{array}{ll}x& \text{ if }x\ge 0\\ -x& \text{ if }x<0\end{array}\right\}$

Case 1: x ≥ 0

If x is positive or zero, then |x| = x.

Therefore, |x| - x = x - x = 0.

Substituting this into our function: f(x) = 1/√(0) = 1/0, which is undefined.

Conclusion: The function is not defined for any x ≥ 0.

Case 2: x < 0

If x is negative, then |x| = -x.

Therefore, |x| - x = (-x) - x = -2x.

Since x is negative, -2x will be positive. For example, if x = -5, |x| - x = 5 - (-5) = 10.

So, for all x < 0, |x| - x > 0. This satisfies our condition that the expression under the square root is positive.

Substituting into our function: f(x) = 1/√(-2x). Since -2x is always positive, the square root is defined and is a positive number. Therefore, the function is defined.

Conclusion: The function is defined for all x < 0.

Step 2: Combine the Results and State the Domain

From our analysis:

• The function is not defined for x ≥ 0.
• The function is defined for x < 0.

Therefore, the domain is all real numbers less than zero. In interval notation, this is written as (-∞, 0).

Final Answer

The domain of the function $f\left(x\right)=\frac{1}{\sqrt{\left|x\right|-x}}$ is (-∞, 0).

This corresponds to the last option: (-∞, 0).

Related Topics & Formulae

Key Concepts

• Domain of a Function: The set of all possible input values (x-values) for which the function is defined.
• Absolute Value: The distance of a number from zero on the number line, always non-negative. |x| = x if x ≥ 0, and |x| = -x if x < 0.
• Square Root Function: The function √(a) is only defined for a ≥ 0. Furthermore, if it is in the denominator of a fraction, we must also have a ≠ 0.

General Approach for Finding Domain

When finding the domain of a function, look for values that cause:

1. Division by zero.
2. An even root (like a square root) of a negative number.
3. A logarithm of a non-positive number.

Exclude any x-values that result in these conditions.

Useful Formulae

Definition of Absolute Value:

$|x|=\sqrt{x^2}$

Inequality for Square Root in Denominator:

For a function of the form $f\left(x\right)=\frac{1}{\sqrt{g\left(x\right)}}$, the domain is found by solving $g\left(x\right)>0$.