https://r2.erweima.ai/mathbot/image/17267044201783mm3xci6.jpg

The graph provided seems to represent a piecewise function based on the red line segments and points shown. Here's how we can analyze the function step by step:

1. **Left Line Segment**:
   - The first part of the graph (for \( x \leq 1 \)) is a straight line with a negative slope. It starts from approximately \( (x = -2, y = 4) \) and passes through \( (x = 1, y = 0) \).
   - Using these two points, we can calculate the slope \( m \) of the line:
     \[
     m = \frac{0 - 4}{1 - (-2)} = \frac{-4}{3} = -\frac{4}{3}
     \]
   - Using the slope-intercept form \( y = mx + b \), where \( m = -\frac{4}{3} \) and the point \( (1, 0) \) to find \( b \) (the y-intercept):
     \[
     0 = -\frac{4}{3}(1) + b \implies b = \frac{4}{3}
     \]
   - So, for \( x \leq 1 \), the equation of the line is:
     \[
     y = -\frac{4}{3}x + \frac{4}{3}
     \]

2. **Right Segment**:
   - The second part of the graph is a horizontal line at \( y = 2 \) starting from \( x = 2 \) (closed dot at \( (2, 2) \)) and ending at \( x = 4 \) (open dot at \( (4, 2) \)).
   - Thus, for \( 2 \leq x < 4 \), the equation is:
     \[
     y = 2
     \]

### The piecewise function:

\[
f(x) =
\begin{cases}
-\frac{4}{3}x + \frac{4}{3}, & \text{for } x \leq 1 \\
2, & \text{for } 2 \leq x < 4
\end{cases}
\]

This describes the function based on the graph provided.

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Would you like further details or have any questions?

### Related Questions:
1. How do we find the equation of a line given two points?
2. What does the slope of a line tell us about the graph?
3. How do open and closed dots on a graph affect the function's domain?
4. Can a piecewise function have more than two segments?
5. How does a graph help us visualize discontinuities in a function?

### Tip:
Always ensure the slope of a line segment is calculated carefully by taking two distinct points and applying the slope formula correctly!

Escribe la expresión analítica correspondiente a la función cuya gráfica es:

This solution explains how to derive a piecewise function from a graph that contains linear segments. The problem covers key concepts like slope-intercept form and piecewise functions, with step-by-step guidance on calculating the slope and writing the function analytically.

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