Math Problem Statement

Complete the description of the piecewise function graphed below. Use interval notation to indicate the intervals.

Solution

The problem asks you to describe the piecewise function using interval notation based on the graph provided. Let's analyze the graph and identify the different segments of the piecewise function:

  1. First Interval:

    • The function is at f(x)=3f(x) = 3 for values of xx between -4 and -2, inclusive of -4 but not -2.
    • Interval: [−4,−2)[-4, -2)

  2. Second Interval:

    • The function is at f(x)=5f(x) = 5 for values of xx between -2 and 1, inclusive of -2 but not 1.
    • Interval: [−2,1)[-2, 1)

  3. Third Interval:

    • The function is at f(x)=−4f(x) = -4 for values of xx between 1 and 3, inclusive of 1 but not 3.
    • Interval: [1,3)[1, 3)

Given this information, the piecewise function description would be:

3 & \text{if } x \in [-4, -2) \\ 5 & \text{if } x \in [-2, 1) \\ -4 & \text{if } x \in [1, 3) \end{cases}$$ Would you like more details on how these intervals were determined? Here are some related questions to deepen your understanding: 1. How would you graph a piecewise function if given the intervals and function values? 2. What is the significance of open and closed circles in the graph of a piecewise function? 3. How does the concept of continuity apply to piecewise functions? 4. Can you describe the process of finding the domain of a piecewise function? 5. How would the graph look if we changed the values of $$f(x)$$ for any of the intervals? **Tip:** Always pay attention to whether the circles on the graph are open or closed, as this affects whether an endpoint is included in the interval.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Interval Notation
Graph Analysis

Formulas

Piecewise function intervals based on the graph

Theorems

None explicitly mentioned, but involves basic principles of function definition and interval notation

Suitable Grade Level

Grades 9-12

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