To graph a **piecewise function**, follow these steps:

### 1. **Understand the Function**
   A piecewise function consists of different expressions, each valid over specific intervals of the domain. For example:

   \[
   f(x) =
   \begin{cases}
   2x + 1 & \text{if } x \leq 0 \\
   -x + 3 & \text{if } x > 0
   \end{cases}
   \]
   Here, there are two pieces. For \(x \leq 0\), the function is \(2x + 1\), and for \(x > 0\), it’s \(-x + 3\).

### 2. **Identify the Domain for Each Piece**
   Each expression applies to a specific range of \(x\)-values. Mark these intervals:
   - \(2x + 1\) for \(x \leq 0\)
   - \(-x + 3\) for \(x > 0\)

### 3. **Plot Each Piece Separately**
   - For \(2x + 1\), plot the line for \(x \leq 0\). This is a straight line with slope 2 and y-intercept 1, but only for \(x \leq 0\).
   - For \(-x + 3\), plot this line for \(x > 0\). It has slope -1 and y-intercept 3, but only for \(x > 0\).

### 4. **Use Open or Closed Circles**
   - At the point where the domain changes (in this case at \(x = 0\)), use a **closed circle** if the inequality includes the value (e.g., \(x \leq 0\)), and an **open circle** if it does not (e.g., \(x > 0\)).

   - At \(x = 0\), you would draw:
     - A closed circle on the graph of \(2x + 1\) at \(x = 0\) because the inequality includes 0.
     - An open circle on the graph of \(-x + 3\) at \(x = 0\) because the inequality does not include 0.

### 5. **Check for Continuity**
   - If the endpoints of different pieces match, the function is continuous. If not, there’s a jump discontinuity at that point. In this example, at \(x = 0\), there’s a discontinuity because \(2(0) + 1 = 1\) and \(-0 + 3 = 3\), so the pieces don't meet.

### Example Graph of a Piecewise Function

If you follow these steps for the above function, you'd get two lines: one rising for \(x \leq 0\) and another descending for \(x > 0\), with a discontinuity at \(x = 0\).

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Do you want more details or have any questions?

Here are 5 questions to expand your understanding:
1. How can you identify if a piecewise function is continuous?
2. What is the difference between an open and closed circle on a graph?
3. How would you graph a piecewise function with three or more pieces?
4. How can you determine the slope of each segment in a piecewise function?
5. Can a piecewise function have different types of discontinuities?

**Tip:** Always carefully label your intervals and use correct symbols to avoid errors when graphing piecewise functions!

Learn how to graph piecewise functions with this step-by-step guide. Understand the key algebraic concepts, plot individual pieces, identify domains, and use open or closed circles. Explore the continuity and discontinuity in functions. Suitable for students in Grades 9-12.

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