Piecewise Function Analysis with Absolute Value and Quadratic Equation

Math Problem Statement

Given a piecewise function f(x) = {5, if x <= -3; |x|, if -3 < x <= 1; x^2 - 2, if x > 1}, how do you analyze and graph it?

The piecewise function given in the image is defined as:

5 & \text{if } x \leq -3 \\ |x| & \text{if } -3 < x \leq 1 \\ x^2 - 2 & \text{if } x > 1 \end{cases}$$ ### Explanation of the function: 1. **For $$x \leq -3$$:** The function is constant and equal to 5. 2. **For $$-3 < x \leq 1$$:** The function is $$|x|$$, the absolute value of $$x$$, which is piecewise itself: - $$|x| = x$$ for $$x \geq 0$$ - $$|x| = -x$$ for $$x < 0$$ 3. **For $$x > 1$$:** The function is quadratic, $$f(x) = x^2 - 2$$, which is a parabola shifted downward by 2 units. ### Graphing Insight: - **At $$x = -3$$:** The function jumps from the constant value $$f(x) = 5$$ to $$f(x) = |x| = 3$$. - **From $$x = -3$$ to $$x = 1$$:** The function follows the absolute value graph, which is V-shaped and symmetric around the y-axis. - **At $$x = 1$$:** The function moves to the quadratic part, transitioning smoothly into a parabola starting from $$f(1) = 1^2 - 2 = -1$$. Would you like me to assist with graphing or further analysis? ### Additional Questions: 1. How would you graph the absolute value portion between $$-3$$ and $$1$$? 2. What are the critical points for this piecewise function? 3. How would you compute the limits at the transition points $$x = -3$$ and $$x = 1$$? 4. Can you describe the behavior of the quadratic part $$x^2 - 2$$ as $$x \to \infty$$? 5. How would the function change if the middle part were $$x^2$$ instead of $$|x|$$? ### Tip: When graphing piecewise functions, always carefully consider the transition points where the function changes form!

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Piecewise Functions
Absolute Value
Quadratic Functions

|x| = x if x >= 0, |x| = -x if x < 0
Quadratic formula: ax^2 + bx + c

Properties of Absolute Value
Properties of Quadratic Functions

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