Math Problem Statement
π(π₯) = { βπ₯2 + 8, ππ π₯ < 0 β3π₯ + 7, ππ 0 β€ π₯ < 4 βπ₯ β 4 + 1, ππ 4 β€ π₯
Solution
The function you've provided is a piecewise function, which means that it is defined by different expressions depending on the value of . Let's break it down:
\begin{cases} -x^2 + 8, & \text{if } x < 0 \\ -3x + 7, & \text{if } 0 \leq x < 4 \\ \sqrt{x - 4} + 1, & \text{if } 4 \leq x \end{cases}$$ This function is defined as: 1. **For $$x < 0$$**: The function follows the equation $$f(x) = -x^2 + 8$$, which is a downward-opening parabola. 2. **For $$0 \leq x < 4$$**: The function follows the equation $$f(x) = -3x + 7$$, which is a straight line with slope -3 and y-intercept 7. 3. **For $$4 \leq x$$**: The function follows the equation $$f(x) = \sqrt{x - 4} + 1$$, which represents a square root function shifted 4 units to the right and 1 unit upward. If you need help evaluating the function for a specific $$x$$-value or want to explore its behavior at key points, feel free to ask! Would you like to see a graph of this function, or do you need any other specific details? ### Questions for further exploration: 1. How would the function behave at $$x = 0$$ and $$x = 4$$? 2. Can you explain how to evaluate $$f(x)$$ at a given point in each of the three intervals? 3. What are the potential limits of $$f(x)$$ as $$x \to -\infty$$ and $$x \to \infty$$? 4. What is the derivative of each piece of the function, and how does it change across the intervals? 5. How do you determine the continuity of this piecewise function at the boundaries (i.e., at $$x = 0$$ and $$x = 4$$)? **Tip**: Always check the function values at the boundaries of each piecewise interval to ensure proper continuity and smooth transitions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Quadratic Functions
Linear Functions
Square Root Functions
Formulas
Quadratic Function: f(x) = ax^2 + bx + c
Linear Function: f(x) = mx + b
Square Root Function: f(x) = sqrt(x - a) + b
Theorems
Continuity at a point: lim(xβc) f(x) = f(c)
Derivative rules for piecewise functions
Limit rules for evaluating function behavior
Suitable Grade Level
Grades 9-12
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