The domain of which function does NOT include 2 ?

Options

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

Explanation

To determine which function does not include 2 in its domain, we need to analyze the denominators of the given functions. A function is not defined at a value where its denominator equals zero, as division by zero is undefined.

### Step-by-Step Solution:

1. **Identify the Functions**: We need to check the denominators of the functions provided in the options. Let's denote the functions as:
- Option A: \( f(x) = \frac{1}{x^2 - 2x^3} \)
- Option B: \( g(x) = \frac{x - 2}{x^3 - 2x^3} \)
- Option C: \( h(x) = \frac{2x^2}{x^3 - 2x^2} \)

2. **Check Option A**:
- The denominator is \( x^2 - 2x^3 \).
- Set the denominator to zero:
\[ x^2 - 2x^3 = 0 \]
- Factor out \( x^2 \):
\[ x^2(1 - 2x) = 0 \]
- This gives us \( x^2 = 0 \) or \( 1 - 2x = 0 \).
- Solving these, we find \( x = 0 \) or \( x = \frac{1}{2} \).
- Thus, the domain of this function includes 2.

3. **Check Option B**:
- The denominator is \( x^3 - 2x^3 \) which simplifies to \( -x^3 \).
- Set the denominator to zero:
\[ -x^3 = 0 \]
- This gives \( x = 0 \).
- Thus, the domain of this function also includes 2.

4. **Check Option C**:
- The denominator is \( x^3 - 2x^2 \).
- Set the denominator to zero:
\[ x^3 - 2x^2 = 0 \]
- Factor out \( x^2 \):
\[ x^2(x - 2) = 0 \]
- This gives us \( x^2 = 0 \) or \( x - 2 = 0 \).
- Solving these, we find \( x = 0 \) or \( x = 2 \).
- Thus, the domain of this function does NOT include 2.

5. **Conclusion**: The function that does not include 2 in its domain is option C.

### Final Answer:
The correct answer is option C: \( h(x) = \frac{2x^2}{x^3 - 2x^2} \).