https://file.aiquickdraw.com/mathbot/file/1728563159376n4tsu3i4.jpg

To express the function shown in the graph as a piecewise function, we need to carefully examine the different parts of the graph and the domains over which each part of the function is defined.

### Analyzing the Graph:
1. **Left Horizontal Line Segment:**
   - This is a horizontal line at \( y = 4 \) extending to the left from \( x = -1 \) and continuing indefinitely.
   - There is an open circle at \( x = -1 \), which means the value at \( x = -1 \) is not included in this part of the function.
   - Domain: \( x < -1 \)
   
2. **Right Diagonal Line Segment:**
   - This is a line with a positive slope starting from \( x = 1 \) at \( y = 5 \) and increasing.
   - There is an open circle at \( x = 1 \), so the value at \( x = 1 \) is not included.
   - From the graph, we can calculate the slope between the points \( (1, 5) \) and \( (6, 9) \).
     - Slope \( m = \frac{9 - 5}{6 - 1} = \frac{4}{5} \)
     - Equation of the line: \( y - 5 = \frac{4}{5}(x - 1) \), simplifying to \( y = \frac{4}{5}x + \frac{1}{5} \).
   - Domain: \( x > 1 \)

### Piecewise Function:
The function can be expressed as:
\[
f(x) =
\begin{cases} 
4 & \text{if } x < -1, \\
\frac{4}{5}x + \frac{1}{5} & \text{if } x > 1.
\end{cases}
\]

Let me know if you'd like further details, or if you have any questions!

### Relative Questions:
1. How do you calculate the slope of a line segment on a graph?
2. What does an open circle on a graph represent in terms of function values?
3. How can piecewise functions represent real-world situations?
4. How do you determine the equation of a line from two points?
5. What is the significance of the domain restrictions in piecewise functions?

**Tip:** When working with piecewise functions, always check the domain carefully to see where each part of the function is valid.

Express the function graphed on the axes below as a piecewise function.

Learn how to express a graphed function as a piecewise function. This problem involves analyzing a graph with different segments and writing equations for each part. Includes concepts like slope, linear equations, and domain restrictions. Suitable for high school students in Grades 9-12.

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