Math Problem Statement
Is f defined at x = 2? Is f continuous at x = 2?
Solution
Let's analyze the problem in the image to answer the questions.
Given Information:
The function is defined piecewise as follows:
\begin{cases} x^2 - 3, & -1 \leq x < 0 \\ 4x, & 0 \leq x < 1 \\ 6, & x = 1 \\ -3x + 7, & 1 < x < 2 \\ 3, & x = 2 \\ 1, & 2 < x < 3 \end{cases}$$ We are asked two questions: 1. **Is $$f$$ defined at $$x = 2$$?** From the piecewise definition, we can see that for $$x = 2$$, $$f(x) = 3$$. Hence, **$$f$$ is defined at $$x = 2$$**. 2. **Is $$f$$ continuous at $$x = 2$$?** To check continuity at $$x = 2$$, we need to verify three things: - $$f(2)$$ is defined (it is, $$f(2) = 3$$). - The left-hand limit $$\lim_{x \to 2^-} f(x)$$. - The right-hand limit $$\lim_{x \to 2^+} f(x)$$. From the piecewise function, for $$1 < x < 2$$, $$f(x) = -3x + 7$$. As $$x \to 2^-$$, $$f(x) = -3(2) + 7 = 1$$. For $$2 < x < 3$$, $$f(x) = 1$$, so $$\lim_{x \to 2^+} f(x) = 1$$. Since both the left-hand and right-hand limits are equal to 1, but $$f(2) = 3$$, the function is **not continuous at $$x = 2$$**. ### Conclusion: - $$f$$ is defined at $$x = 2$$ (answer to part a: Yes). - $$f$$ is not continuous at $$x = 2$$ (answer to part b: No). Would you like further clarification on any part of this solution? Here are five related questions: 1. How can you determine the continuity of a function at any given point? 2. What role do limits play in determining continuity? 3. What does it mean for a function to have a jump discontinuity? 4. Can a function be defined but not continuous at a point? 5. How can you determine if a function is continuous on a given interval? **Tip:** When checking for continuity at a point, always confirm the function's value at the point and evaluate the left-hand and right-hand limits separately.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Limits
Formulas
Limit of a function at a point
Continuity condition: f(a) = lim_{x -> a^-} f(x) = lim_{x -> a^+} f(x)
Theorems
Limit Definition of Continuity
Suitable Grade Level
College Level (Calculus)