) f(x) = x + |x| graph and domain and range

To graph the function \( f(x) = x + |x| \), we first need to analyze its behavior for different values of \( x \).

### 1. Analysis of the function:

- **For \( x \geq 0 \):**
  \[
  f(x) = x + |x| = x + x = 2x
  \]
  So, for \( x \geq 0 \), the function becomes \( f(x) = 2x \).

- **For \( x < 0 \):**
  \[
  f(x) = x + |x| = x - x = 0
  \]
  So, for \( x < 0 \), the function becomes \( f(x) = 0 \).

### 2. Graph of the function:
The graph will consist of two parts:
- A line with slope 2 passing through the origin for \( x \geq 0 \).
- A constant value of 0 for \( x < 0 \).

The graph will look like a ray starting from \( (0, 0) \) and going upwards with slope 2 for \( x \geq 0 \), and a horizontal line at \( f(x) = 0 \) for \( x < 0 \).

### 3. Domain of the function:
The domain is the set of all possible \( x \)-values for which the function is defined. Since the function includes both \( x \geq 0 \) and \( x < 0 \), the domain is:
\[
\text{Domain} = (-\infty, \infty)
\]

### 4. Range of the function:
The range is the set of all possible \( y \)-values (or \( f(x) \)) the function can take. From the behavior of the function:
- For \( x < 0 \), \( f(x) = 0 \).
- For \( x \geq 0 \), \( f(x) = 2x \), meaning \( f(x) \geq 0 \).

So, the range is:
\[
\text{Range} = [0, \infty)
\]

### Summary:
- **Graph**: Piecewise function with a ray from the origin and a constant zero line.
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( [0, \infty) \)

Would you like a more detailed explanation or the graph? 

Here are 5 related questions:
1. How do we graph the absolute value function?
2. What happens when we subtract \( |x| \) instead of adding it?
3. Can we generalize the graph of \( f(x) = x + |x| \) to other functions with absolute values?
4. How does the derivative of \( f(x) = x + |x| \) behave for different \( x \)?
5. What is the impact of adding constants to absolute value functions on their graph?

**Tip**: When analyzing piecewise functions, break them down into intervals and understand how the function behaves in each interval.

f(x) = x + |x| graph and domain and range

Understand how to graph the piecewise function f(x) = x + |x| and analyze its domain and range. This detailed explanation covers the behavior of absolute value functions, including step-by-step graphing instructions, domain, and range determination. Suitable for students in Grades 9-12.

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