Math Problem Statement
Analyze the properties of the piecewise function: f(x) = {cos x, x ≤ 0; x^2 + 1, 0 < x < 1; x, x ≥ 1}
Solution
The function shown in the image is a piecewise-defined function with different expressions for different intervals of :
\cos x, & x \leq 0 \\ x^2 + 1, & 0 < x < 1 \\ x, & x \geq 1 \end{cases}$$ Let's analyze each part of this piecewise function: 1. **For $$x \leq 0$$**: The function is defined as $$f(x) = \cos x$$. This part uses the cosine function, which is defined and continuous for all real $$x$$. 2. **For $$0 < x < 1$$**: The function is defined as $$f(x) = x^2 + 1$$. This is a quadratic expression and is continuous and differentiable over this open interval. 3. **For $$x \geq 1$$**: The function is defined as $$f(x) = x$$, which is a linear function and is also continuous and differentiable. This function may require evaluation for continuity and differentiability at the boundaries (i.e., at $$x = 0$$ and $$x = 1$$), as piecewise functions often need checks at the points where the definition changes. Would you like me to analyze the continuity and differentiability at these boundaries, or do you have other questions? ### Related Questions 1. How do you test for continuity at points where a piecewise function changes? 2. What does it mean for a function to be differentiable at a boundary point? 3. How would the behavior of $$f(x)$$ differ if we changed one of the intervals? 4. What are the values of $$f(0)$$ and $$f(1)$$ for this function? 5. How can we graph a piecewise function like this one? ### Tip For piecewise functions, always check continuity and differentiability at the points where each piece connects. This ensures the function behaves smoothly across its entire domain.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Differentiability
Quadratic Functions
Trigonometric Functions
Formulas
f(x) = cos(x)
f(x) = x^2 + 1
f(x) = x
Theorems
Continuity at boundary points
Differentiability at boundary points
Suitable Grade Level
Grades 11-12
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