Math Problem Statement
Find the values for f(-1), f(1), and limits for a piecewise function with three parts: 8x^2 if x < -1, 8x if -1 ≤ x < 1, and 8/x if x ≥ 1.
Solution
Let's analyze the function and the corresponding questions step by step based on the image.
The given function is a piecewise function:
8x^2 & \text{if } x < -1 \\ 8x & \text{if } -1 \leq x < 1 \\ \frac{8}{x} & \text{if } x \geq 1 \end{cases}$$ You are asked to find the following values: 1. **$$f(-1) = -8$$**: This is correct. For $$x = -1$$, we use the second part of the function $$f(x) = 8x$$, so: $$f(-1) = 8(-1) = -8$$ 2. **$$f(1) = 8$$**: This is correct. For $$x = 1$$, we use the third part of the function $$f(x) = \frac{8}{x}$$, so: $$f(1) = \frac{8}{1} = 8$$ 3. **$$\lim_{x \to -1^-} f(x) = -\infty$$**: This value is marked incorrect. Let's evaluate: - As $$x \to -1$$ from the left, we use the first part of the function $$f(x) = 8x^2$$. - As $$x \to -1$$, we get: $$\lim_{x \to -1^-} 8x^2 = 8(-1)^2 = 8$$ So, the correct answer should be 8, not $$-\infty$$. The marked answer of $$-\infty$$ is indeed incorrect. 4. **$$\lim_{x \to -1^+} f(x) = -8$$**: This is correct. As $$x \to -1^+$$, we use the second part of the function $$f(x) = 8x$$, and: $$\lim_{x \to -1^+} 8x = 8(-1) = -8$$ 5. **$$\lim_{x \to 1^-} f(x) = 8$$**: This is correct. As $$x \to 1^-$$, we use the second part of the function $$f(x) = 8x$$, so: $$\lim_{x \to 1^-} 8x = 8(1) = 8$$ 6. **$$\lim_{x \to 1^+} f(x) = 8$$**: This is correct. As $$x \to 1^+$$, we use the third part of the function $$f(x) = \frac{8}{x}$$, and: $$\lim_{x \to 1^+} \frac{8}{x} = \frac{8}{1} = 8$$ --- Would you like further details on any of these solutions? Here are five related questions for further exploration: 1. What happens to the limit $$\lim_{x \to -1^-} f(x)$$ for functions with non-quadratic terms? 2. How can piecewise continuity at points like $$x = -1$$ and $$x = 1$$ be verified mathematically? 3. What would the graph of this piecewise function look like? 4. How does the behavior of the function change for large values of $$x$$? 5. How would you approach finding the derivative of such a piecewise function? **Tip**: When evaluating limits in piecewise functions, always check the behavior from both sides of the point in question to verify if the limit exists and is finite.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Continuity
Formulas
f(x) = 8x^2 if x < -1
f(x) = 8x if -1 ≤ x < 1
f(x) = 8/x if x ≥ 1
Theorems
Limit definition of a function
One-sided limits
Suitable Grade Level
Grades 11-12
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