The ''panvoid'' of a function is defined as the sum of the integers that do not fall within the domain of the function. All of the following functions have equal panvoids EXCEPT

Options

A. `f(x)=(3x)/(x^(2)-4)`
B. `f(x)=(3-x)/(x^(3)-x)`
C. `f(x)=(3x)/(3x^(2)-27)`
D. `f(x)=(2-x)/(x^(2)-x)`

Explanation

To solve the problem, we need to determine the "panvoid" of each function given in the options. The panvoid is defined as the sum of the integers that do not fall within the domain of the function, which typically occurs when the denominator of the function is equal to zero.

Let's analyze each option step by step.

### Step 1: Analyze Option A
The function is given as \( f(x) = \frac{1}{x^2 - 4} \).

1. **Find the denominator**:
\[ x^2 - 4 \]
2. **Set the denominator to zero**:
\[ x^2 - 4 = 0 \implies x^2 = 4 \implies x = 2 \text{ or } x = -2 \]
3. **Integers not in the domain**:
The integers that do not fall within the domain are \( 2 \) and \( -2 \).
4. **Calculate the panvoid**:
\[ \text{panvoid} = 2 + (-2) = 0 \]

### Step 2: Analyze Option B
The function is given as \( f(x) = \frac{1}{x^3 - x} \).

1. **Find the denominator**:
\[ x^3 - x \]
2. **Factor the denominator**:
\[ x(x^2 - 1) = x(x - 1)(x + 1) \]
3. **Set the denominator to zero**:
\[ x = 0, x = 1, x = -1 \]
4. **Integers not in the domain**:
The integers that do not fall within the domain are \( 0, 1, -1 \).
5. **Calculate the panvoid**:
\[ \text{panvoid} = 0 + 1 + (-1) = 0 \]

### Step 3: Analyze Option C
The function is given as \( f(x) = \frac{1}{3x^2 - 27} \).

1. **Find the denominator**:
\[ 3x^2 - 27 \]
2. **Set the denominator to zero**:
\[ 3x^2 - 27 = 0 \implies 3x^2 = 27 \implies x^2 = 9 \implies x = 3 \text{ or } x = -3 \]
3. **Integers not in the domain**:
The integers that do not fall within the domain are \( 3 \) and \( -3 \).
4. **Calculate the panvoid**:
\[ \text{panvoid} = 3 + (-3) = 0 \]

### Step 4: Analyze Option D
The function is given as \( f(x) = \frac{2 - x}{x^2 - x} \).

1. **Find the denominator**:
\[ x^2 - x \]
2. **Factor the denominator**:
\[ x(x - 1) \]
3. **Set the denominator to zero**:
\[ x = 0 \text{ or } x = 1 \]
4. **Integers not in the domain**:
The integers that do not fall within the domain are \( 0 \) and \( 1 \).
5. **Calculate the panvoid**:
\[ \text{panvoid} = 0 + 1 = 1 \]

### Conclusion
Now we can summarize the panvoids for each option:
- Option A: panvoid = 0
- Option B: panvoid = 0
- Option C: panvoid = 0
- Option D: panvoid = 1

Thus, all of the following functions have equal panvoids **EXCEPT** option D.