If `f(x)=3x+|x|`, then the value of `f(3x)+f(-x)-f(x)` is:

Options

A. `3(x+|x|)^(2)`
B. `3(x+|x|)`
C. `(x-|x|)^(3)`
D. None of these

Answer

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

### Step-by-Step Solution:

1. **Calculate \( f(3x) \)**:
\[ f(3x) = 3(3x) + |3x| = 9x + |3x| \]
Since \( 3x \) is positive when \( x \geq 0 \), we have \( |3x| = 3x \). Thus,
\[ f(3x) = 9x + 3x = 12x \quad \text{(for } x \geq 0\text{)} \]
For \( x < 0 \), \( |3x| = -3x \), so:
\[ f(3x) = 9x - 3x = 6x \quad \text{(for } x < 0\text{)} \]

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

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

4. **Combine the results**:
Now we need to evaluate \( f(3x) + f(-x) - f(x) \):
- For \( x \geq 0 \):
\[ f(3x) + f(-x) - f(x) = (12x) + (-3x + x) - (3x + x) \]
\[ = 12x - 3x + x - 3x - x = 12x - 7x = 5x \]

- For \( x < 0 \):
\[ f(3x) + f(-x) - f(x) = (6x) + (-3x + (-x)) - (3x - x) \]
\[ = 6x - 3x - x - 3x + x = 6x - 6x = 0 \]

5. **Final Result**:
Thus, the value of \( f(3x) + f(-x) - f(x) \) is:
- \( 5x \) for \( x \geq 0 \)
- \( 0 \) for \( x < 0 \)

### Summary:
The expression \( f(3x) + f(-x) - f(x) \) evaluates to:
- \( 5x \) for \( x \geq 0 \)
- \( 0 \) for \( x < 0 \)