Select the correct option

Options

A. `f(x) = e^(3 - 2x). Tan^(2)x + |x| - tan x ` is an even function
B. `f(x) = h(x) + h(-x) + x^(2) cot x` is an even function
C. `f(x) = {{:(x", " x in Q),(-x", " x !in Q):}` is an odd function
D. f(x) = g(x) + g(-x) + |sgn(x)| is neither even nor odd function

Answer

To solve the problem of determining the correct option regarding the nature of the given functions, we will analyze each option step-by-step.

### Step 1: Analyze Option A
**Statement:** The function is an even function.

**Definition of Even Function:** A function \( f(x) \) is even if \( f(-x) = f(x) \).

**Calculation:**
1. Calculate \( f(-x) \).
- Given \( f(x) = e^{3 - 2x} + 2x + 10^2 - x + |x| - 10(-x) \).
- Substitute \(-x\) into the function:
\[ f(-x) = e^{3 - 2(-x)} + 2(-x) + 10^2 - (-x) + |-x| - 10(-(-x)) \]
\[ = e^{3 + 2x} - 2x + 10^2 + x + |x| + 10x \]
\[ = e^{3 + 2x} + 8x + 10^2 + |x| \]

2. Compare \( f(-x) \) with \( f(x) \):
- Since \( e^{3 + 2x} \neq e^{3 - 2x} \), the two expressions are not equal.
- Therefore, **Option A is incorrect**.

### Step 2: Analyze Option B
**Statement:** The function is an odd function.

**Definition of Odd Function:** A function \( f(x) \) is odd if \( f(-x) = -f(x) \).

**Calculation:**
1. We already calculated \( f(-x) \) in the previous step.
2. Now calculate \(-f(x)\):
\[ -f(x) = -\left(e^{3 - 2x} + 2x + 10^2 - x + |x| - 10(-x)\right) \]
\[ = -e^{3 - 2x} - 2x - 10^2 + x - |x| + 10x \]
\[ = -e^{3 - 2x} + 8x - 10^2 - |x| \]

3. Compare \( f(-x) \) with \(-f(x)\):
- The terms do not match, hence \( f(-x) \neq -f(x) \).
- Therefore, **Option B is incorrect**.

### Step 3: Analyze Option C
**Statement:** The function is neither even nor odd.

**Analysis:**
- From the previous steps, we found that \( f(x) \) does not satisfy the conditions for being either even or odd.
- Thus, **Option C is correct**.

### Step 4: Analyze Option D
**Statement:** The function is neither even nor odd.

**Analysis:**
- This statement is consistent with our findings in Step 3.
- Therefore, **Option D is also correct**.

### Conclusion
Both Options C and D state that the function is neither even nor odd. However, since the question asks to select the correct option, we conclude that **Option D is the most appropriate choice**.

### Final Answer
**Correct Option:** D