Find the domain of f(x)=1/(sqrt(|[|x| -1]|-5).

where [ ] is G.I.F.

To find the domain of the function \( f(x) = \frac{1}{\sqrt{||x| - 1| - 5}} \), we need to ensure that the expression inside the square root is positive, as the square root function is only defined for non-negative values. Additionally, since the function is in the denominator, it cannot be zero. Let's break this down step by step.

Step 1: Identify the conditions for the square root

For the square root to be valid, the expression must be greater than zero:

• We need \( ||x| - 1| - 5 > 0 \).

Step 2: Solve the inequality

This inequality can be rewritten as:

• \( ||x| - 1| > 5 \)

This means that the absolute value \( |x| - 1 \) must be either greater than 5 or less than -5. We will consider both cases.

Case 1: \( |x| - 1 > 5 \)

Solving this gives:

• \( |x| > 6 \)
• This implies \( x < -6 \) or \( x > 6 \).

Case 2: \( |x| - 1 < -5 \)

Now, solving this gives:

• \( |x| < -4 \)

This case is not possible because the absolute value cannot be negative. Therefore, we disregard this case.

Step 3: Combine the results

The only valid solutions come from Case 1, which leads us to the conclusion that the domain of \( f(x) \) consists of values of \( x \) that are either less than -6 or greater than 6. In interval notation, this is expressed as:

• \( (-\infty, -6) \cup (6, \infty) \)

Final Domain Statement

Thus, the domain of the function \( f(x) = \frac{1}{\sqrt{||x| - 1| - 5}} \) is all real numbers except those between -6 and 6, inclusive. In simpler terms, you can plug in any number as long as it's either less than -6 or greater than 6.