`f(x)=1/([abs(x-2}]+[abs(x-10)]-8)` where `[*]` denotes the greatest integer function.

Answer

To solve the problem, we need to analyze the function \( f(x) = \frac{1}{[\lvert x-2 \rvert] + [\lvert x-10 \rvert] - 8} \), where \([\cdot]\) denotes the greatest integer function (also known as the floor function).

### Step 1: Understand the components of the function
The function consists of two absolute value expressions: \(\lvert x-2 \rvert\) and \(\lvert x-10 \rvert\). The greatest integer function will take the values of these absolute expressions and round them down to the nearest integer.

### Step 2: Determine the intervals for the greatest integer function
The expressions \(\lvert x-2 \rvert\) and \(\lvert x-10 \rvert\) change their values at \(x = 2\) and \(x = 10\). Therefore, we will analyze the function in the following intervals:
1. \( (-\infty, 2) \)
2. \( [2, 10) \)
3. \( [10, \infty) \)

### Step 3: Analyze each interval

#### Interval 1: \( x < 2 \)
- Here, \(\lvert x-2 \rvert = 2 - x\) and \(\lvert x-10 \rvert = 10 - x\).
- Thus, \([\lvert x-2 \rvert] = [2 - x]\) and \([\lvert x-10 \rvert] = [10 - x]\).
- The sum becomes \([\lvert x-2 \rvert] + [\lvert x-10 \rvert] = [2 - x] + [10 - x]\).
- For \(x < 2\), both \(2 - x\) and \(10 - x\) are positive, and we can find specific values for \(x\) to evaluate the floor functions.

#### Interval 2: \( 2 \leq x < 10 \)
- Here, \(\lvert x-2 \rvert = x - 2\) and \(\lvert x-10 \rvert = 10 - x\).
- Thus, \([\lvert x-2 \rvert] = [x - 2]\) and \([\lvert x-10 \rvert] = [10 - x]\).
- The sum becomes \([\lvert x-2 \rvert] + [\lvert x-10 \rvert] = [x - 2] + [10 - x]\).
- We will analyze specific values in this interval.

#### Interval 3: \( x \geq 10 \)
- Here, \(\lvert x-2 \rvert = x - 2\) and \(\lvert x-10 \rvert = x - 10\).
- Thus, \([\lvert x-2 \rvert] = [x - 2]\) and \([\lvert x-10 \rvert] = [x - 10]\).
- The sum becomes \([\lvert x-2 \rvert] + [\lvert x-10 \rvert] = [x - 2] + [x - 10]\).
- Again, we will analyze specific values in this interval.

### Step 4: Find the domain of \( f(x) \)
We need to find when the denominator \([\lvert x-2 \rvert] + [\lvert x-10 \rvert] - 8\) is not equal to zero. This means we need to find values of \(x\) such that:

\[ [\lvert x-2 \rvert] + [\lvert x-10 \rvert] \neq 8 \]

### Step 5: Identify the points where the function is undefined
By evaluating the intervals and checking where the sum equals 8, we can find the points where \(f(x)\) is undefined.

### Final Step: Write the domain
After analyzing the intervals and finding the points where the function is undefined, we can conclude the domain of \(f(x)\).