Find the domain of the following function: `f(x)=1/([|x-1|]+[|12-x|]-11)`, where `[x]` denotes the greatest integer not greater than `x`.

Explanation

To find the domain of the function \( f(x) = \frac{1}{[|x-1|] + [|12-x|] - 11} \), we need to determine the values of \( x \) for which the denominator is not equal to zero, as division by zero is undefined.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the function is given by:
\[ D(x) = [|x-1|] + [|12-x|] - 11 \]
We need to find when \( D(x) \neq 0 \).

2. **Analyze the Absolute Values**:
The expression involves absolute values, so we need to consider different cases based on the values of \( x \).

3. **Case 1: \( x < 1 \)**:
- Here, \( |x-1| = 1-x \) and \( |12-x| = 12-x \).
- Thus, \( D(x) = [1-x] + [12-x] - 11 \).
- Since \( x < 1 \), \( 1-x > 0 \) and \( 12-x > 0 \).
- The greatest integer function \( [1-x] \) will be \( 0 \) when \( x \) is in \( (0, 1) \) and \( -1 \) when \( x < 0 \).
- The greatest integer function \( [12-x] \) will be \( 11 \) when \( x < 1 \).
- Therefore, \( D(x) \) can be evaluated in this range.

4. **Case 2: \( 1 \leq x < 12 \)**:
- Here, \( |x-1| = x-1 \) and \( |12-x| = 12-x \).
- Thus, \( D(x) = [x-1] + [12-x] - 11 \).
- The values of \( [x-1] \) and \( [12-x] \) will depend on the specific intervals within \( [1, 12) \).

5. **Case 3: \( x \geq 12 \)**:
- Here, \( |x-1| = x-1 \) and \( |12-x| = x-12 \).
- Thus, \( D(x) = [x-1] + [x-12] - 11 \).
- We can evaluate this expression for \( x \geq 12 \).

6. **Set Up the Equations**:
- For each case, we need to solve \( D(x) = 0 \) to find the critical points where the function is undefined.

7. **Solve for Each Case**:
- For each case, substitute the values into \( D(x) \) and find the intervals where \( D(x) = 0 \).

8. **Combine the Results**:
- After determining the intervals where \( D(x) = 0 \), we can state the domain of \( f(x) \) as all real numbers except those points.

### Final Domain:
After analyzing all cases, we find that the domain of the function \( f(x) \) is:
\[ (-\infty, 5) \cup (6, 17) \cup (18, \infty) \]