https://file.aiquickdraw.com/mathbot/file/172685867064335n22g7x.jpg

Let's analyze the function \( f(x) \) defined piecewise:

\[
f(x) =
\begin{cases}
|x| & \text{for } x < 2 \\
-x + 4 & \text{for } x \geq 2
\end{cases}
\]

### Steps to graph the function:

1. **For \( x < 2 \), the function is \( f(x) = |x| \):**
   - The absolute value function \( |x| \) is V-shaped, with a vertex at \( x = 0 \).
   - For \( x < 2 \), plot the points:
     - \( f(-3) = 3 \)
     - \( f(-2) = 2 \)
     - \( f(0) = 0 \)
     - \( f(1) = 1 \)
     - As \( x \) approaches 2 from the left, \( f(1.5) = 1.5 \) and \( f(2) = 2 \), but this is the boundary for the next function piece.
  
2. **For \( x \geq 2 \), the function is \( f(x) = -x + 4 \):**
   - This is a linear function with a slope of -1 and a y-intercept of 4.
   - Plot points for \( x \geq 2 \):
     - At \( x = 2 \), \( f(2) = 2 \) (continuous from the previous piece).
     - \( f(3) = -3 + 4 = 1 \)
     - \( f(4) = -4 + 4 = 0 \)
     - \( f(5) = -5 + 4 = -1 \)
   
Now graph the two parts:

- For \( x < 2 \), the function is the absolute value graph, with the point \( (2, 2) \) included.
- For \( x \geq 2 \), graph the line \( -x + 4 \), starting from \( (2, 2) \).

Would you like further clarification or help with plotting? Here are some questions for further practice:

1. What is the domain and range of this piecewise function?
2. Is the function continuous at \( x = 2 \)?
3. How would the graph change if the linear part was \( -x + 5 \)?
4. How do we define the slope for \( f(x) = -x + 4 \)?
5. What is the behavior of the absolute value function as \( x \to -\infty \)?

**Tip:** When graphing piecewise functions, always check for continuity and slope at the boundary points.

Graph the piecewise function f(x) = |x| for x < 2 and f(x) = -x + 4 for x ≥ 2

Learn how to graph the piecewise function where f(x) = |x| for x < 2 and f(x) = -x + 4 for x ≥ 2. This covers key concepts of absolute value functions, linear equations, and graphing piecewise functions. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation