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.