Math Problem Statement
how to graph piecewise
Solution
To graph a piecewise function, follow these steps:
1. Understand the Function
A piecewise function consists of different expressions, each valid over specific intervals of the domain. For example:
\begin{cases} 2x + 1 & \text{if } x \leq 0 \\ -x + 3 & \text{if } x > 0 \end{cases}$$ Here, there are two pieces. For $$x \leq 0$$, the function is $$2x + 1$$, and for $$x > 0$$, it’s $$-x + 3$$. ### 2. **Identify the Domain for Each Piece** Each expression applies to a specific range of $$x$$-values. Mark these intervals: - $$2x + 1$$ for $$x \leq 0$$ - $$-x + 3$$ for $$x > 0$$ ### 3. **Plot Each Piece Separately** - For $$2x + 1$$, plot the line for $$x \leq 0$$. This is a straight line with slope 2 and y-intercept 1, but only for $$x \leq 0$$. - For $$-x + 3$$, plot this line for $$x > 0$$. It has slope -1 and y-intercept 3, but only for $$x > 0$$. ### 4. **Use Open or Closed Circles** - At the point where the domain changes (in this case at $$x = 0$$), use a **closed circle** if the inequality includes the value (e.g., $$x \leq 0$$), and an **open circle** if it does not (e.g., $$x > 0$$). - At $$x = 0$$, you would draw: - A closed circle on the graph of $$2x + 1$$ at $$x = 0$$ because the inequality includes 0. - An open circle on the graph of $$-x + 3$$ at $$x = 0$$ because the inequality does not include 0. ### 5. **Check for Continuity** - If the endpoints of different pieces match, the function is continuous. If not, there’s a jump discontinuity at that point. In this example, at $$x = 0$$, there’s a discontinuity because $$2(0) + 1 = 1$$ and $$-0 + 3 = 3$$, so the pieces don't meet. ### Example Graph of a Piecewise Function If you follow these steps for the above function, you'd get two lines: one rising for $$x \leq 0$$ and another descending for $$x > 0$$, with a discontinuity at $$x = 0$$. --- Do you want more details or have any questions? Here are 5 questions to expand your understanding: 1. How can you identify if a piecewise function is continuous? 2. What is the difference between an open and closed circle on a graph? 3. How would you graph a piecewise function with three or more pieces? 4. How can you determine the slope of each segment in a piecewise function? 5. Can a piecewise function have different types of discontinuities? **Tip:** Always carefully label your intervals and use correct symbols to avoid errors when graphing piecewise functions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Piecewise Functions
Graphing
Formulas
f(x) = 2x + 1 for x ≤ 0
f(x) = -x + 3 for x > 0
Theorems
Continuity and Discontinuity in Functions
Suitable Grade Level
Grades 9-12