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To express the function shown in the graph as a piecewise function, we need to analyze the behavior of the function on different intervals of the x-axis.

From the graph:

- For \( x < 0 \), the function is a horizontal line at \( y = -4 \). This section includes a solid point at \( (-1, -4) \), which means the function is defined as \( y = -4 \) for \( x = -1 \).
- For \( x = 0 \), there is no defined value at \( x = 0 \), as the function has an open circle at \( (0, 5) \).
- For \( x > 0 \), the function is a horizontal line at \( y = 5 \), but the function is not defined at \( x = 0 \) because of the open circle.

### Piecewise function:

\[
f(x) = 
\begin{cases} 
-4 & \text{for } x \leq -1 \\
5 & \text{for } x > 0 
\end{cases}
\]

This represents the behavior of the function in different parts of the graph.

Would you like further clarification or details on how the function was derived?

Here are 5 related questions for further exploration:

1. How would the piecewise function change if there were a defined value at \( x = 0 \)?
2. What is the significance of the open and closed circles in defining piecewise functions?
3. Can a piecewise function have more than two intervals? What would that look like graphically?
4. How would you calculate the domain and range of this piecewise function?
5. What are common applications of piecewise functions in real-life problems?

**Tip:** Always check for open and closed circles in a graph, as they determine whether the function is defined or undefined at certain points.

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

Learn how to express a function as a piecewise function based on its graph. This problem covers the core concepts of piecewise functions and graph interpretation, including open and closed points, and function behavior across intervals. Suitable for students in Grades 9-11.

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