https://file.aiquickdraw.com/mathbot/file/17266195193426te3061z.jpg

The graph in the image represents a piecewise function. Here's how the function can be expressed based on the graph:

1. From \( x = -\infty \) to \( x = 1 \), the graph shows a horizontal line at \( y = 3 \). However, at \( x = 1 \), there is an open circle, indicating that the function is not defined at \( x = 1 \).
   
   Thus, for \( x < 1 \), the function is:
   \[
   f(x) = 3 \quad \text{for} \quad x < 1
   \]

2. At \( x = 1 \), the function has a closed dot at \( y = 5 \), indicating that the value of the function at this point is 5.

   Therefore:
   \[
   f(x) = 5 \quad \text{for} \quad x = 1
   \]

3. For \( x > 1 \), the graph continues with a horizontal line at \( y = 3 \), meaning the function returns to the value 3 for all \( x > 1 \).

   Hence, for \( x > 1 \):
   \[
   f(x) = 3 \quad \text{for} \quad x > 1
   \]

### Complete Piecewise Function:
\[
f(x) = \begin{cases} 
3 & \text{for} \, x < 1 \\
5 & \text{for} \, x = 1 \\
3 & \text{for} \, x > 1
\end{cases}
\]

Would you like more details or have any questions about this function?

Here are 5 questions to deepen your understanding:
1. How does an open or closed circle on a graph affect the piecewise function?
2. Can piecewise functions have more than two conditions?
3. How would you represent a non-continuous function using piecewise notation?
4. Why is the function at \( x = 1 \) different from its neighboring values?
5. Can piecewise functions represent real-life situations? If so, how?

**Tip:** When identifying a piecewise function from a graph, pay close attention to open and closed circles, as they define the function's domain at specific 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 a graph. This problem involves interpreting key concepts like open and closed circles and translating them into algebraic form. Suitable for students in Grades 9-12.

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