https://file.aiquickdraw.com/mathbot/file/1727311727546pbc42kd3.jpg

To determine the piecewise function \( f(x) \) based on the graph, I will analyze the two line segments and their corresponding domains. Each line segment represents a portion of the function.

### 1. Left Line Segment (From \( x = -5 \) to \( x = 1 \)):

- **Endpoints**: The line starts at \( (-5, 6) \) and ends at \( (1, -2) \).
- **Slope Calculation**:
  \[
  \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 6}{1 - (-5)} = \frac{-8}{6} = -\frac{4}{3}
  \]
- **Equation of the Line**: 
  Use the point-slope form \( y - y_1 = m(x - x_1) \) with \( m = -\frac{4}{3} \) and point \( (-5, 6) \):
  \[
  y - 6 = -\frac{4}{3}(x + 5)
  \]
  Simplifying this:
  \[
  y - 6 = -\frac{4}{3}x - \frac{20}{3}
  \]
  \[
  y = -\frac{4}{3}x + \frac{-20 + 18}{3} = -\frac{4}{3}x - \frac{2}{3}
  \]
  Thus, for \( -5 \leq x \leq 1 \), the equation is:
  \[
  f(x) = -\frac{4}{3}x - \frac{2}{3}
  \]

### 2. Right Line Segment (From \( x = 1 \) to \( x = 5 \)):

- **Endpoints**: The line starts at \( (1, -2) \) and ends at \( (5, 4) \).
- **Slope Calculation**:
  \[
  \text{slope} = \frac{4 - (-2)}{5 - 1} = \frac{6}{4} = \frac{3}{2}
  \]
- **Equation of the Line**:
  Use the point-slope form \( y - y_1 = m(x - x_1) \) with \( m = \frac{3}{2} \) and point \( (1, -2) \):
  \[
  y + 2 = \frac{3}{2}(x - 1)
  \]
  Simplifying this:
  \[
  y + 2 = \frac{3}{2}x - \frac{3}{2}
  \]
  \[
  y = \frac{3}{2}x - \frac{3}{2} - 2 = \frac{3}{2}x - \frac{7}{2}
  \]
  Thus, for \( 1 < x \leq 5 \), the equation is:
  \[
  f(x) = \frac{3}{2}x - \frac{7}{2}
  \]

### Final Piecewise Function:

The function \( f(x) \) is given by:
\[
f(x) = \begin{cases} 
-\frac{4}{3}x - \frac{2}{3} & \text{for } -5 \leq x \leq 1 \\
\frac{3}{2}x - \frac{7}{2} & \text{for } 1 < x \leq 5 
\end{cases}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you determine the slope between two points?
2. What is the point-slope form of a linear equation?
3. How would the function change if the slope of one of the segments were different?
4. What is the significance of open and closed circles in piecewise functions?
5. How can you graph a piecewise function from its algebraic definition?

**Tip:** When writing piecewise functions, always ensure the domain matches the behavior in the graph.

Learn how to determine a piecewise function from a graph with two linear segments. This step-by-step solution involves calculating slopes and writing equations for each segment. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation