`f:R->R` is defined as `f(x)=2x+|x|` then `f(3x)-f(-x)-4x=`

Options

A. a) `f(x)`
B. b) `-f(x)`
C. c) `f(-x)`
D. d) `2f(x)`

Explanation

To solve the problem, we need to evaluate the expression \( f(3x) - f(-x) - 4x \) where the function \( f(x) = 2x + |x| \).

### Step-by-Step Solution:

1. **Define the function**:
\[ f(x) = 2x + |x| \]

2. **Calculate \( f(3x) \)**:
\[ f(3x) = 2(3x) + |3x| = 6x + 3|x| \]

3. **Calculate \( f(-x) \)**:
\[ f(-x) = 2(-x) + |-x| = -2x + |x| \]

4. **Substitute \( f(3x) \) and \( f(-x) \) into the expression**:
\[ f(3x) - f(-x) - 4x = (6x + 3|x|) - (-2x + |x|) - 4x \]

5. **Simplify the expression**:
- First, distribute the negative sign:
\[ = 6x + 3|x| + 2x - |x| - 4x \]
- Combine like terms:
\[ = (6x + 2x - 4x) + (3|x| - |x|) = 4x + 2|x| \]

6. **Factor out the common term**:
\[ = 2(2x + |x|) \]

7. **Recognize that \( 2(2x + |x|) \) is \( 2f(x) \)**:
\[ = 2f(x) \]

### Final Result:
Thus, we find that:
\[ f(3x) - f(-x) - 4x = 2f(x) \]

### Conclusion:
The answer is \( 2f(x) \).