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

Explanation

To find all values of \( f(x) \) for which \( f(x) = x + \sqrt{x^2} \), we can analyze the expression step by step.

### Step 1: Understand the expression \( \sqrt{x^2} \)

The expression \( \sqrt{x^2} \) can be simplified based on the value of \( x \):

- If \( x \geq 0 \) (i.e., \( x \) is non-negative), then \( \sqrt{x^2} = x \).
- If \( x < 0 \) (i.e., \( x \) is negative), then \( \sqrt{x^2} = -x \).

### Step 2: Analyze the cases based on the sign of \( x \)

Now, we will consider two cases based on the value of \( x \):

#### Case 1: \( x \geq 0 \)

For \( x \geq 0 \):
\[ f(x) = x + \sqrt{x^2} = x + x = 2x \]

#### Case 2: \( x < 0 \)

For \( x < 0 \):
\[ f(x) = x + \sqrt{x^2} = x - x = 0 \]

### Step 3: Combine the results

Now we can summarize the results from both cases:

- For \( x \geq 0 \), \( f(x) = 2x \)
- For \( x < 0 \), \( f(x) = 0 \)

Thus, we can express \( f(x) \) as:
\[ f(x) = \begin{cases} 2x & \text{if } x \geq 0 \\ 0 & \text{if } x < 0 \end{cases} \]

### Final Result

The final values of \( f(x) \) are:
\[ f(x) = 2x \text{ for } x \geq 0 \quad \text{and} \quad f(x) = 0 \text{ for } x < 0 \]

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