Math Problem Statement
Graph the piecewise function: f(x) = {2x, for x < 0; x + 5, for x >= 0}. Choose the correct graph.
Solution
We are given the piecewise function:
\begin{cases} 2x, & \text{for } x < 0 \\ x + 5, & \text{for } x \geq 0 \end{cases}$$ We need to identify the correct graph for this function. Let's analyze the behavior of the function on each interval: ### 1. For $$x < 0$$: The function is $$f(x) = 2x$$, which is a straight line with slope 2 passing through the origin. It applies only for values of $$x$$ that are negative (left side of the y-axis). ### 2. For $$x \geq 0$$: The function is $$f(x) = x + 5$$, which is another straight line with a slope of 1 and a y-intercept of 5. This applies for values of $$x$$ that are greater than or equal to 0 (right side of the y-axis), and includes the point $$(0, 5)$$. ### Key Features to Check: - For $$x < 0$$, the line should follow $$2x$$, which means a steep line passing through the origin but only on the left of the y-axis. - For $$x \geq 0$$, the line should start at $$(0, 5)$$ and have a slope of 1 (a gentler slope). ### Now, let’s examine the options: - **Option A**: This shows a line starting at $$(0, 5)$$ on the right but does not match the slope on the left side. - **Option B**: This graph has the correct lines for both the left and right sections. - **Option C**: Incorrect for the left side since the slope does not match. - **Option D**: The slope on the right side is incorrect. ### Conclusion: The correct graph is **Option B**, as it correctly reflects the behavior of both parts of the piecewise function. Would you like further explanation or details on any part of the function? Here are five follow-up questions: 1. How do you find the slope of a line given its equation? 2. What is the significance of the y-intercept in a linear function? 3. How do you determine where a piecewise function changes its rule? 4. What happens to the graph of a function when you change the slope? 5. How does continuity apply to piecewise functions? **Tip**: When graphing piecewise functions, always check the slope and y-intercept for each segment, and note where the rule changes.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Formulas
f(x) = 2x for x < 0
f(x) = x + 5 for x >= 0
Theorems
Slope-Intercept Form
Suitable Grade Level
Grades 9-11