 Select the algebraic definition for the piecewise function graph.


Piecewise function graphed


To select the algebraic definition of a **piecewise function** from its graph, we need to:

1. **Identify the intervals**: Look at where the graph changes behavior (e.g., discontinuities or different slopes).
2. **Define the function on each interval**: Write an expression for each piece of the graph on its respective interval.
3. **Incorporate boundaries**: Use appropriate inequality symbols (like \( \leq \) or \( < \)) to specify which intervals the expressions are valid on.

For example, if a graph shows two linear segments:  
- The first segment goes from \( x = -\infty \) to \( x = 0 \) with a slope of 2,  
- The second segment goes from \( x = 0 \) to \( x = \infty \) with a slope of -1,

The algebraic definition would be written like this:

\[
f(x) = 
\begin{cases} 
2x + 1 & \text{for } x \leq 0 \\
-x + 2 & \text{for } x > 0
\end{cases}
\]

Would you like to upload the graph of the piecewise function so I can help define it specifically?

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Here are 5 questions to expand understanding:
1. What is the process of identifying intervals in a piecewise function graph?
2. How do you determine the slope of a piece from its graph?
3. What is the importance of inequalities in the definition of piecewise functions?
4. How can we determine if a function is continuous at the boundary of its pieces?
5. What real-world scenarios can be modeled using piecewise functions?

**Tip**: Always check for **open vs closed circles** on the graph to see whether an interval includes or excludes a boundary point.

Select the algebraic definition for the piecewise function graph.

Learn how to select the algebraic definition for a piecewise function based on its graph. This guide covers identifying function intervals, defining expressions for each interval, and applying inequalities for boundaries. Ideal for Grades 10-12 students studying advanced algebra.

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