Graph the piecewise function below by dragging the endpoints of the lines to the correct locations, then select whether the endpoints are included in the graph or not.
f
(
x
)
=
{
x
+
1
if
x
<
−
2
−
2
x
−
3
if
x
≥
−
2


Let's break down the given piecewise function to understand how to graph it:

\[
f(x) = 
\begin{cases} 
x + 1 & \text{if } x < -2 \\
-2x - 3 & \text{if } x \geq -2
\end{cases}
\]

### Step 1: Analyze the pieces of the function
- **For \( x + 1 \) when \( x < -2 \):**
  This is a linear function with a slope of 1 and a y-intercept of 1. However, it only applies for \( x < -2 \), so the domain is restricted.
  
- **For \( -2x - 3 \) when \( x \geq -2 \):**
  This is also a linear function, but with a slope of -2 and a y-intercept of -3. It applies only for \( x \geq -2 \), including the point where \( x = -2 \).

### Step 2: Find the endpoints
- **For \( x = -2 \),**
  - The first part (\( x + 1 \)) does **not include** \( x = -2 \), because the condition is \( x < -2 \).
  - The second part (\( -2x - 3 \)) **does include** \( x = -2 \), because the condition is \( x \geq -2 \).
  
To check the value of the function at \( x = -2 \):
  - For \( x + 1 \), the point approaches \( (-2, -1) \) but is **not included**.
  - For \( -2x - 3 \), at \( x = -2 \), the point is \( (-2, 1) \) and is **included**.

### Step 3: Graph the two lines

1. **For \( x + 1 \), graph the line starting just to the left of \( x = -2 \) and moving leftward.** 
   - The line will approach \( (-2, -1) \), but since \( x = -2 \) is not included in this piece, place an open circle at \( (-2, -1) \).
   - From there, continue drawing the line with slope 1, so for every unit increase in \( x \), increase \( y \) by 1.

2. **For \( -2x - 3 \), graph the line starting at \( (-2, 1) \) and moving rightward.**
   - The line will start at \( (-2, 1) \), and since \( x = -2 \) is included, place a solid dot at \( (-2, 1) \).
   - From there, continue drawing the line with slope -2, so for every unit increase in \( x \), decrease \( y \) by 2.

### Final Plot Characteristics:
- An open circle at \( (-2, -1) \) from the first piece.
- A solid dot at \( (-2, 1) \) from the second piece.
- The first line slopes upward (positive slope) and the second slopes downward (negative slope).

Would you like me to provide an actual graph or go into more details on graphing techniques?

---

Here are 5 related questions for practice:
1. How do you determine whether to include an endpoint in a piecewise function graph?
2. What are the slopes of each piece of the function, and how do they affect the graph?
3. How do you find the points where the pieces of a piecewise function meet?
4. What happens if two pieces of a piecewise function meet at the same point but with different rules for inclusion?
5. How do you shift or stretch a linear function in a graph?

**Tip:** Always pay close attention to whether inequalities in piecewise functions are strict (\( < \) or \( > \)) or inclusive (\( \leq \) or \( \geq \))—this affects whether endpoints are open or closed.

Graph the piecewise function below by dragging the endpoints of the lines to the correct locations, then select whether the endpoints are included in the graph or not.
f(x) = { x + 1 if x < -2, -2x - 3 if x ≥ -2 }

Learn how to graph a piecewise function defined by two linear equations with different domain restrictions. This step-by-step solution covers graphing techniques, analyzing endpoints, and determining whether points are included or excluded. Ideal for Grades 9-11.

Math Problem Statement

Graph the piecewise function below by dragging the endpoints of the lines to the correct locations, then select whether the endpoints are included in the graph or not. f ( x )

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation