Math Problem Statement
Solve
Solution
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Problem Breakdown
The problem appears to involve sketching a graph based on a piecewise function defined as:
x^2 & \text{if } x < 1 \\ x + 1 & \text{if } 1 \leq x \leq 3 \\ 5 - x & \text{if } x > 3 \end{cases}$$ ### Solution Steps 1. **Sketching for $$x < 1$$:** - Here, the function is $$f(x) = x^2$$. - This is a parabola opening upwards, but we are only interested in the part where $$x < 1$$. - Specifically, at $$x = 1$$, $$f(1) = 1^2 = 1$$, so the parabola reaches the point $$(1, 1)$$, but is not included (since $$x < 1$$). 2. **Sketching for $$1 \leq x \leq 3$$:** - The function is $$f(x) = x + 1$$. - This is a straight line with a slope of 1, starting at $$(1, 2)$$ and ending at $$(3, 4)$$. 3. **Sketching for $$x > 3$$:** - The function is $$f(x) = 5 - x$$. - This is a downward-sloping line, starting at $$(3, 2)$$ (but not included) and extending downward. ### Final Graph: - The parabola starts from negative infinity and stops just before $$x = 1$$ with an open circle at $$(1, 1)$$. - The line from $$(1, 2)$$ to $$(3, 4)$$ is drawn with closed circles at both ends. - Another line starts just after $$x = 3$$ (open circle at $$(3, 2)$$) and extends downward. This provides a continuous graph of the piecewise function. ### Next Steps Do you want further details on how to plot this graph? Or perhaps some clarification on specific steps? Here are some related questions to consider: 1. What is the derivative of each piece of the function, and how does it affect the graph? 2. What happens to the graph if we change the domain of each piecewise section? 3. How would the graph look if we reflected it across the y-axis? 4. Can we express this piecewise function in terms of a single absolute value function? 5. How do we find the intersection points if another function is added to this graph? **Tip:** When graphing piecewise functions, always check the boundary points to ensure the function's continuity or note any discontinuities.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Graphing
Formulas
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Theorems
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Suitable Grade Level
High School