Math Problem Statement
Graph the piecewise function: f(x) = { 3x + 12 for x ≤ -2, -x + 13 for x > 4 }.
Solution
You have provided a piecewise function:
3x + 12 & \text{for } x \leq -2 \\ -x + 13 & \text{for } x > 4 \end{cases}$$ To graph this, we need to handle each part of the function separately: 1. **For $$x \leq -2$$:** The equation is $$3x + 12$$. At $$x = -2$$, the point is $$f(-2) = 3(-2) + 12 = 6$$, so we have the point $$(-2, 6)$$. The slope is 3, so the line will rise 3 units for every 1 unit moved to the right. The line continues for all $$x \leq -2$$. 2. **For $$x > 4$$:** The equation is $$-x + 13$$. At $$x = 4$$, the point is $$f(4) = -(4) + 13 = 9$$, but since this is for $$x > 4$$, we begin just after $$x = 4$$. The slope is -1, meaning the line decreases by 1 unit for every 1 unit moved to the right. To graph it: - Draw the line $$y = 3x + 12$$ starting from $$(-2, 6)$$ and extending to the left. - Draw the line $$y = -x + 13$$ starting just after $$(4, 9)$$ and extending to the right. Would you like me to generate a visual graph based on these instructions? --- Here are 5 questions to further explore: 1. How do you determine the slope from an equation of a line? 2. What is the domain of the given piecewise function? 3. How does the graph change if we modify the breakpoints $$x = -2$$ and $$x = 4$$? 4. What is the range of the piecewise function based on the graph? 5. Can you explain the behavior of the function as $$x \to -\infty$$ and $$x \to \infty$$? **Tip:** Always check the domain restrictions carefully when dealing with piecewise functions, as they define where each part of the function is applicable.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Graphing
Formulas
y = 3x + 12
y = -x + 13
Theorems
Linear Function Properties
Slope-Intercept Form
Suitable Grade Level
Grades 9-11