Math Problem Statement
Graph the following function on the axes provided: f(x) = { -2x + 4 for x < -1 ; -5x + 19 for x > 3 }.
Solution
To graph the given piecewise function , we need to analyze the two linear equations provided for their respective domains.
Function Definition:
\begin{cases} -2x + 4 & \text{for } x < -1 \\ -5x + 19 & \text{for } x > 3 \end{cases}$$ ### Steps to Graph: 1. **Graph $$f(x) = -2x + 4$$ for $$x < -1$$:** - Find a few points on this line within the domain $$x < -1$$. - Example points: - For $$x = -2$$: $$f(-2) = -2(-2) + 4 = 8$$. - For $$x = -3$$: $$f(-3) = -2(-3) + 4 = 10$$. - Plot these points and extend the line to the left of $$x = -1$$. 2. **Graph $$f(x) = -5x + 19$$ for $$x > 3$$:** - Find a few points on this line within the domain $$x > 3$$. - Example points: - For $$x = 4$$: $$f(4) = -5(4) + 19 = -1$$. - For $$x = 5$$: $$f(5) = -5(5) + 19 = -6$$. - Plot these points and extend the line to the right of $$x = 3$$. 3. **Verify the endpoints:** - Check the values of the function at the boundaries $$x = -1$$ and $$x = 3$$, if needed. - The function does not include these boundary points, so it should have open circles at these boundaries if you plot them. ### Graphing Tip: To represent the function accurately, be sure to clearly show the open circles at $$x = -1$$ and $$x = 3$$ as these values are not included in the function definition. If you need further assistance with graphing or any details, feel free to ask! --- **Relative Questions:** 1. How do you graph a piecewise function with more than two segments? 2. What are the key differences in graphing continuous vs. discontinuous piecewise functions? 3. How do you find the slope and y-intercept of each segment in a piecewise function? 4. Can a piecewise function have a constant segment, and how would you graph it? 5. How do you determine if a piecewise function is continuous at a boundary point? **Tip:** When graphing piecewise functions, always check the inclusion (closed circle) or exclusion (open circle) of boundary points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing Linear Equations
Slope-Intercept Form
Formulas
Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept
Piecewise Function: f(x) defined by multiple sub-functions over different intervals
Theorems
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Suitable Grade Level
High School (Grades 9-12)