Find all values of f(x) for which f(x) `=x+sqrt(x^2)`

Explanation

To solve the problem, we need to analyze the function \( f(x) = x + \sqrt{x^2} \).

### Step 1: Rewrite the function
The expression \( \sqrt{x^2} \) can be simplified. The square root of \( x^2 \) is the absolute value of \( x \), denoted as \( |x| \). Therefore, we can rewrite the function as:
\[ f(x) = x + |x| \]

### Step 2: Consider cases for \( |x| \)
The absolute value function \( |x| \) has two cases based on the value of \( x \):
1. When \( x \geq 0 \), \( |x| = x \)
2. When \( x < 0 \), \( |x| = -x \)

### Step 3: Evaluate the function for each case
**Case 1: \( x \geq 0 \)**
- Here, \( |x| = x \)
- Thus, the function becomes:
\[ f(x) = x + x = 2x \]

**Case 2: \( x < 0 \)**
- Here, \( |x| = -x \)
- Thus, the function becomes:
\[ f(x) = x - x = 0 \]

### Step 4: Combine the results
From the evaluations in both cases, we can summarize the function \( f(x) \) as:
\[ f(x) = \begin{cases} 2x & \text{if } x \geq 0 \\ 0 & \text{if } x < 0 \end{cases} \]

### Step 5: Determine the domain of \( f(x) \)
The function \( f(x) \) is defined for all real numbers \( x \). However, the expression \( \sqrt{x^2} \) is non-negative for all \( x \), which means \( f(x) \) is valid for all \( x \).

### Final Result
Thus, the function \( f(x) \) is defined for all \( x \in \mathbb{R} \) and can be expressed piecewise as shown above.