If `f(x)=xandg(x)=|x|`, then what is `(f+g)(x)` equal to ?

Options

A. 0 for all `x""inR`
B. 2x for all `x""inR`
C. `{{:(2x","" for "xge0),(0","" for "xlt0):}`
D. `{{:(0","" for "xge0),(2x","" for "xlt0):}`

Explanation

To find \((f+g)(x)\) where \(f(x) = x\) and \(g(x) = |x|\), we will follow these steps:

1. **Define the Functions**:
- We have \(f(x) = x\).
- We have \(g(x) = |x|\).

2. **Combine the Functions**:
- The sum of the functions \(f\) and \(g\) is given by:
\[ (f+g)(x) = f(x) + g(x) = x + |x| \]

3. **Evaluate the Absolute Value**:
- The absolute value function \(|x|\) behaves differently based on the sign of \(x\):
- If \(x \geq 0\), then \(|x| = x\).
- If \(x < 0\), then \(|x| = -x\).

4. **Case Analysis**:
- **Case 1**: If \(x \geq 0\):
\[ (f+g)(x) = x + |x| = x + x = 2x \]
- **Case 2**: If \(x < 0\):
\[ (f+g)(x) = x + |x| = x - x = 0 \]

5. **Final Result**:
- Therefore, we can summarize the result as:
\[ (f+g)(x) = \begin{cases} 2x & \text{if } x \geq 0 \\ 0 & \text{if } x < 0 \end{cases} \]