Math Problem Statement
Graph the piecewise function f(x) = { x-1 for x < -3, -5 for -3 ≤ x ≤ 1, 4x for x > 1 }, and choose the correct graph.
Solution
Let's analyze the piecewise function :
\begin{cases} x - 1 & \text{for } x < -3, \\ -5 & \text{for } -3 \leq x \leq 1, \\ 4x & \text{for } x > 1. \end{cases}$$ ### 1. First interval: $$x - 1$$ for $$x < -3$$ This is a line with a slope of 1 and a y-intercept of -1. For $$x < -3$$, this line will be graphed only to the left of $$x = -3$$. ### 2. Second interval: $$-5$$ for $$-3 \leq x \leq 1$$ This is a constant function, which means it will be a horizontal line at $$y = -5$$ for $$x$$ between -3 and 1, inclusive of both endpoints. ### 3. Third interval: $$4x$$ for $$x > 1$$ This is another linear function with a slope of 4 and a y-intercept of 0. It will only be graphed to the right of $$x = 1$$. ### Now, let's match the options: - **Option A**: This graph appears to show a correct piecewise function where: - $$x - 1$$ for $$x < -3$$ is sloping downward (incorrect slope). - The constant function $$y = -5$$ between $$-3 \leq x \leq 1$$ is shown correctly. - $$4x$$ for $$x > 1$$ appears correct. - **Option B**: Seems to match the correct behavior: - The line $$x - 1$$ for $$x < -3$$ has the correct slope. - The constant function $$y = -5$$ is in place for $$-3 \leq x \leq 1$$. - The line $$4x$$ starts correctly from $$x = 1$$. - **Options C and D**: Both of these contain inaccuracies in how the piecewise function is presented, such as incorrect slopes or misplacements of the constant segment. Thus, **Option B** is the correct graph for this piecewise function. Would you like more details on how each interval was graphed? Also, here are some related questions for further understanding: 1. How do you graph a piecewise function with multiple intervals? 2. What is the significance of open and closed dots in piecewise graphs? 3. How do you determine the slope of a linear function? 4. Why is there a break in the graph of a piecewise function? 5. How does the constant function behave in piecewise functions? **Tip**: Always check the domain restrictions for each part of a piecewise function to ensure you graph only within the specified intervals.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Functions
Constant Functions
Formulas
f(x) = x - 1 for x < -3
f(x) = -5 for -3 ≤ x ≤ 1
f(x) = 4x for x > 1
Theorems
Domain and Range in Piecewise Functions
Slope-Intercept Form of a Line
Suitable Grade Level
Grades 9-12