https://file.aiquickdraw.com/mathbot/file/1725935128414wzi6h4ac.jpg

The piecewise function given is:

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

### To sketch the graph of this function:

1. **For \( x \leq -1 \)**:
   - The function is constant, \( f(x) = 0 \), which means the graph is a horizontal line at \( y = 0 \) for all values \( x \leq -1 \). 

2. **For \( -1 < x \leq 2 \)**:
   - The function is \( f(x) = x + 1 \), a linear function with slope 1 and y-intercept at \( (0, 1) \). It is only valid in the interval \( -1 < x \leq 2 \), so we plot this line from just after \( x = -1 \) to \( x = 2 \). 
   - At \( x = -1 \), the function has a hole since the interval is open at \( -1 \).
   - At \( x = 2 \), the function is defined, so we place a closed point there with \( f(2) = 2 + 1 = 3 \).

3. **For \( x > 2 \)**:
   - The function is constant again, \( f(x) = 0 \), meaning the graph is a horizontal line at \( y = 0 \) for all \( x > 2 \).

### Steps to plot:
- Draw a horizontal line on the x-axis (at \( y = 0 \)) for \( x \leq -1 \) and \( x > 2 \).
- Draw a line with slope 1 between \( -1 < x \leq 2 \), starting at \( (-1, 0) \) (open circle) and ending at \( (2, 3) \) (closed circle).
  
Would you like more details or a further breakdown of any step?

---

Here are 5 related questions to consider:
1. How does changing the slope in the middle interval affect the graph?
2. What happens if the piecewise function is continuous at all points?
3. How would the graph change if the intervals for the linear function were modified?
4. Can this piecewise function be transformed into another form, like an absolute value function?
5. How would this graph change if there were a different constant function on each side of the middle interval?

**Tip**: Always check if the endpoints in a piecewise function are open or closed to ensure accurate graph representation!

Sketch a graph of f(x) = { 0 if x ≤ -1, x + 1 if -1 < x ≤ 2, 0 if x > 2 }.

Learn how to sketch the graph of a piecewise function defined by three segments: constant at y = 0, linear segment x + 1, and another constant at y = 0. Detailed instructions on plotting each interval, with open and closed points, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation