https://file.aiquickdraw.com/mathbot/file/17261449845654teu6gei.jpg

You have provided a piecewise function:

\[
f(x) = \begin{cases} 
3x + 12 & \text{for } x \leq -2 \\
-x + 13 & \text{for } x > 4 
\end{cases}
\]

To graph this, we need to handle each part of the function separately:

1. **For \( x \leq -2 \):**  
   The equation is \( 3x + 12 \).  
   At \( x = -2 \), the point is \( f(-2) = 3(-2) + 12 = 6 \), so we have the point \( (-2, 6) \).  
   The slope is 3, so the line will rise 3 units for every 1 unit moved to the right. The line continues for all \( x \leq -2 \).

2. **For \( x > 4 \):**  
   The equation is \( -x + 13 \).  
   At \( x = 4 \), the point is \( f(4) = -(4) + 13 = 9 \), but since this is for \( x > 4 \), we begin just after \( x = 4 \).  
   The slope is -1, meaning the line decreases by 1 unit for every 1 unit moved to the right.

To graph it:

- Draw the line \( y = 3x + 12 \) starting from \( (-2, 6) \) and extending to the left.
- Draw the line \( y = -x + 13 \) starting just after \( (4, 9) \) and extending to the right.

Would you like me to generate a visual graph based on these instructions?

---

Here are 5 questions to further explore:

1. How do you determine the slope from an equation of a line?
2. What is the domain of the given piecewise function?
3. How does the graph change if we modify the breakpoints \( x = -2 \) and \( x = 4 \)?
4. What is the range of the piecewise function based on the graph?
5. Can you explain the behavior of the function as \( x \to -\infty \) and \( x \to \infty \)?

**Tip:** Always check the domain restrictions carefully when dealing with piecewise functions, as they define where each part of the function is applicable.

Graph the piecewise function: f(x) = { 3x + 12 for x ≤ -2, -x + 13 for x > 4 }.

This problem involves graphing a piecewise function with two linear equations. For x ≤ -2, the function follows y = 3x + 12, and for x > 4, it follows y = -x + 13. Explore key concepts in linear graphing and understand how to handle domain restrictions in piecewise functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation