Math Problem Statement
Graph each of the following piecewise functions. Identify key characteristics.
Solution
The worksheet contains piecewise functions, and the task is to graph them and identify key characteristics. Let's analyze them:
1st Piecewise Function:
\begin{cases} x + 5, & \text{if } x < -2 \\ -4, & \text{if } x \geq -2 \end{cases}$$ - For $$x < -2$$, the function follows the equation $$f(x) = x + 5$$, which is a linear function. - For $$x \geq -2$$, the function is constant $$f(x) = -4$$. **Key Details:** - The line $$f(x) = x + 5$$ starts at $$x = -\infty$$ and stops at $$x = -2$$ (open circle at $$(-2,3)$$). - At $$x = -2$$, the function jumps to $$f(x) = -4$$ (solid point at $$(-2, -4)$$) and remains constant for $$x \geq -2$$. --- ### **2nd Piecewise Function:** $$f(x) = \begin{cases} 2x + 1, & \text{if } x < 1 \\ -2x + 3, & \text{if } x \geq 1 \end{cases}$$ - For $$x < 1$$, the function follows the linear equation $$f(x) = 2x + 1$$. - For $$x \geq 1$$, it follows the linear equation $$f(x) = -2x + 3$$. **Key Details:** - The first function starts at $$x = -\infty$$ and stops at $$x = 1$$ (open circle at $$(1,3)$$). - The second function begins at $$x = 1$$ and continues (solid point at $$(1,1)$$), following a different slope. --- Would you like a step-by-step explanation for graphing? Here are some related questions: 1. How do you determine continuity in piecewise functions? 2. What is the significance of open and closed circles in graphing? 3. How do you find the domain and range of a piecewise function? 4. How can you check if a piecewise function is differentiable? 5. What are real-world applications of piecewise functions? **Tip:** Always check the function value at the transition point to see if the graph is continuous! 🚀Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Continuity
Formulas
f(x) = x + 5 for x < -2
f(x) = -4 for x ≥ -2
f(x) = 2x + 1 for x < 1
f(x) = -2x + 3 for x ≥ 1
Theorems
Piecewise Function Behavior
Continuity at Transition Points
Suitable Grade Level
Grades 9-12
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