If `f : R rarr R` be defined as `f(x) = 2x + |x|`, then `f(2x) + f(-x) - f(x)` is equal to

Options

A. `-2x`
B. `2x`
C. `-2|x|`
D. `2|x|`

Answer

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

### Step 1: Calculate \( f(2x) \)
Using the definition of the function:
\[ f(2x) = 2(2x) + |2x| = 4x + |2x| \]
Since \( |2x| = 2|x| \), we can rewrite this as:
\[ f(2x) = 4x + 2|x| \]

### Step 2: Calculate \( f(-x) \)
Using the definition of the function:
\[ f(-x) = 2(-x) + |-x| = -2x + |x| \]

### Step 3: Calculate \( f(x) \)
Using the definition of the function:
\[ f(x) = 2x + |x| \]

### Step 4: Substitute into the expression
Now we substitute \( f(2x) \), \( f(-x) \), and \( f(x) \) into the expression \( f(2x) + f(-x) - f(x) \):
\[ f(2x) + f(-x) - f(x) = (4x + 2|x|) + (-2x + |x|) - (2x + |x|) \]

### Step 5: Simplify the expression
Now we simplify the expression:
\[ = 4x + 2|x| - 2x + |x| - 2x - |x| \]
Combining like terms:
\[ = (4x - 2x - 2x) + (2|x| + |x| - |x|) \]
This simplifies to:
\[ = 0 + 2|x| = 2|x| \]

### Final Result
Thus, the expression \( f(2x) + f(-x) - f(x) \) is equal to:
\[ \boxed{2|x|} \]