Piecewise Function Analysis with Continuity and Differentiability

Math Problem Statement

What is the behavior of the piecewise function f(x), defined as f(x) = x + 2 for x ≤ -1, f(x) = -x^2 + 2x for -1 < x < 1, and f(x) = 1/x for x > 1?

The uploaded image shows a piecewise-defined function $f(x)$ given by:

\begin{cases} x + 2 & \text{if } x \leq -1 \\ -x^2 + 2x & \text{if } -1 < x < 1 \\ \frac{1}{x} & \text{if } x > 1 \end{cases}$$ Would you like to analyze the properties of this function? I can help with the following: 1. **Domain and range** 2. **Continuity at points** like $$x = -1$$ and $$x = 1$$ 3. **First derivative** for checking differentiability or finding critical points 4. **Graph sketch** 5. **Limits** as $$x \to -1$$, $$x \to 1$$, and $$x \to \infty$$ --- Do you have any specific analysis in mind or shall we start with continuity and domain? Here are five questions that can expand on this: 1. What are the points where the function might not be continuous? 2. Is the function differentiable at $$x = -1$$ and $$x = 1$$? 3. What are the limits of $$f(x)$$ as $$x$$ approaches the boundaries? 4. How do we find the critical points of the quadratic part? 5. What is the behavior of $$f(x) = \frac{1}{x}$$ as $$x \to \infty$$? **Tip:** Piecewise functions are often tested for **continuity** and **differentiability** at points where the rules change.

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Piecewise Functions
Continuity
Differentiability
Quadratic Functions
Reciprocal Functions

f(x) = x + 2 for x ≤ -1
f(x) = -x^2 + 2x for -1 < x < 1
f(x) = 1/x for x > 1

Continuity Theorem
Differentiability Theorem
Limit Theorem

Grades 11-12 or early college

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Understanding Piecewise Functions with Boundaries and Continuity

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Piecewise Function with Continuity and Differentiability Analysis