Math Problem Statement

Analyze the continuity and differentiability of the piecewise function provided in the image.

Solution

The piecewise function provided in the image is:

\begin{cases} 2x + 1, & \text{if } x \leq -1 \\ 3x, & \text{if } -1 < x < 1 \text{ and at } x = -1 \text{ and } x = 1 \\ 2x - 1, & \text{if } x \geq 1 \end{cases}$$ Let's analyze this function step-by-step: ### 1. Continuity Check To check the continuity at the points where the definition of the function changes, specifically at $$x = -1$$ and $$x = 1$$, we need to ensure that the left-hand limit (LHL), right-hand limit (RHL), and the function value at those points are equal. #### At $$x = -1$$: - Left-hand limit: $$\lim\limits_{x \to -1^-} f(x) = 2(-1) + 1 = -2 + 1 = -1$$ - Right-hand limit: $$\lim\limits_{x \to -1^+} f(x) = 3(-1) = -3$$ - $$f(-1)$$ according to the second case is $$3(-1) = -3$$ Since LHL $$\neq$$ RHL, the function is **not continuous** at $$x = -1$$. #### At $$x = 1$$: - Left-hand limit: $$\lim\limits_{x \to 1^-} f(x) = 3(1) = 3$$ - Right-hand limit: $$\lim\limits_{x \to 1^+} f(x) = 2(1) - 1 = 2 - 1 = 1$$ - $$f(1)$$ according to the second case is $$3(1) = 3$$ Since LHL $$\neq$$ RHL, the function is **not continuous** at $$x = 1$$. ### 2. Differentiability Check For a function to be differentiable at a point, it must be continuous there, and the left-hand derivative must equal the right-hand derivative. Since we already found that the function is not continuous at $$x = -1$$ and $$x = 1$$, the function is also **not differentiable** at these points. Would you like more details on any part of the solution, or do you have any specific questions? ### 5 Relative Questions: 1. How would you determine if a piecewise function is differentiable at a given point? 2. Can you give an example of a piecewise function that is both continuous and differentiable at all points? 3. What is the significance of continuity in real-world applications of piecewise functions? 4. How can you graph a piecewise function to visualize its continuity and differentiability? 5. What are the general conditions for a function to be continuous and differentiable? ### Tip: When dealing with piecewise functions, always carefully analyze the points where the definition changes to check for continuity and differentiability.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Continuity
Differentiability
Limits

Formulas

Left-hand limit and Right-hand limit formulas
Conditions for continuity
Conditions for differentiability

Theorems

Limit Definition of Continuity
Differentiability Implies Continuity

Suitable Grade Level

Grades 10-12