Find the values of x for which the following function is defined: `f(x)=sqrt((1)/(|x-2|-(x-2)))`

Explanation

To find the values of \( x \) for which the function \( f(x) = \sqrt{\frac{1}{|x-2| - (x-2)}} \) is defined, we need to ensure that the expression inside the square root is non-negative and that the denominator is not zero.

### Step 1: Analyze the absolute value expression
The absolute value \( |x-2| \) can be expressed in two cases:
1. When \( x \geq 2 \): \( |x-2| = x-2 \)
2. When \( x < 2 \): \( |x-2| = -(x-2) = 2-x \)

### Step 2: Case 1 - When \( x \geq 2 \)
For \( x \geq 2 \):
\[ f(x) = \sqrt{\frac{1}{(x-2) - (x-2)}} = \sqrt{\frac{1}{0}} \]
This expression is undefined because the denominator is zero.

### Step 3: Case 2 - When \( x < 2 \)
For \( x < 2 \):
\[ f(x) = \sqrt{\frac{1}{(2-x) - (x-2)}} = \sqrt{\frac{1}{(2-x) - (x-2)}} \]
Simplifying the denominator:
\[ (2-x) - (x-2) = 2 - x - x + 2 = 4 - 2x \]
Thus, we have:
\[ f(x) = \sqrt{\frac{1}{4 - 2x}} \]
For this expression to be defined, \( 4 - 2x \) must be greater than 0:
\[ 4 - 2x > 0 \implies 4 > 2x \implies 2 > x \]
This means \( x < 2 \).

### Step 4: Combine the results
From the analysis:
- For \( x \geq 2 \), the function is undefined.
- For \( x < 2 \), the function is defined as long as \( 4 - 2x > 0 \).

Thus, the function \( f(x) \) is defined for:
\[ x < 2 \]

### Final Answer
The values of \( x \) for which the function \( f(x) \) is defined are:
\[ (-\infty, 2) \]