https://file.aiquickdraw.com/mathbot/file/1724213230638trjsph60.jpg

The graph of the function f in the figure resembles the letter M. Define f(x) piecewise.

To define the function \( f(x) \) based on the provided graph, we will break it down into the different linear segments that comprise the "M" shape. Let's analyze the graph:

### Analysis of the Graph
1. The function starts from \( x = -2 \) and rises to \( x = -1 \).
2. From \( x = -1 \) to \( x = 0 \), the function descends.
3. From \( x = 0 \) to \( x = 1 \), the function ascends again.
4. From \( x = 1 \) to \( x = 2 \), the function descends back to the x-axis.

We will write the equation of each line segment and then combine them to form a piecewise function.

### Define the Segments

1. **Segment from \( x = -2 \) to \( x = -1 \):**
   - The points on this line segment are \( (-2, 0) \) and \( (-1, 2) \).
   - The slope \( m_1 \) of the line is:
     \[
     m_1 = \frac{2 - 0}{-1 - (-2)} = \frac{2}{1} = 2
     \]
   - The equation of this line is:
     \[
     y = 2(x + 2)
     \]
   - Simplifying:
     \[
     f(x) = 2x + 4 \quad \text{for } -2 \leq x \leq -1
     \]

2. **Segment from \( x = -1 \) to \( x = 0 \):**
   - The points are \( (-1, 2) \) and \( (0, 0) \).
   - The slope \( m_2 \) is:
     \[
     m_2 = \frac{0 - 2}{0 - (-1)} = \frac{-2}{1} = -2
     \]
   - The equation is:
     \[
     y = -2x + 2
     \]
   - So:
     \[
     f(x) = -2x + 2 \quad \text{for } -1 \leq x \leq 0
     \]

3. **Segment from \( x = 0 \) to \( x = 1 \):**
   - The points are \( (0, 0) \) and \( (1, 2) \).
   - The slope \( m_3 \) is:
     \[
     m_3 = \frac{2 - 0}{1 - 0} = 2
     \]
   - The equation is:
     \[
     y = 2x
     \]
   - So:
     \[
     f(x) = 2x \quad \text{for } 0 \leq x \leq 1
     \]

4. **Segment from \( x = 1 \) to \( x = 2 \):**
   - The points are \( (1, 2) \) and \( (2, 0) \).
   - The slope \( m_4 \) is:
     \[
     m_4 = \frac{0 - 2}{2 - 1} = \frac{-2}{1} = -2
     \]
   - The equation is:
     \[
     y = -2x + 4
     \]
   - So:
     \[
     f(x) = -2x + 4 \quad \text{for } 1 \leq x \leq 2
     \]

### Piecewise Function Definition
Combining all the segments, the piecewise function \( f(x) \) is:
\[
f(x) =
\begin{cases}
2x + 4 & \text{for } -2 \leq x \leq -1 \\
-2x + 2 & \text{for } -1 \leq x \leq 0 \\
2x & \text{for } 0 \leq x \leq 1 \\
-2x + 4 & \text{for } 1 \leq x \leq 2
\end{cases}
\]

Would you like to go over any specific part in more detail, or do you have any questions?

### Related Questions:
1. How do you determine the slope of a line segment between two points?
2. What are the key differences between continuous and piecewise functions?
3. Can piecewise functions have non-linear segments?
4. How would you represent a piecewise function graphically?
5. What are some common real-world examples of piecewise functions?

### Tip:
When graphing a piecewise function, it's helpful to mark the endpoints of each segment clearly and ensure that the function values at those points match for continuity, if required.

Learn how to define the piecewise function f(x) resembling the 'M' graph shape step-by-step. This problem covers piecewise functions, linear equations, and graphical analysis. 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