Math Problem Statement
Diketahui f(x) = { x^2 - 1, x ≤ -1 ; 2x + 2, x > -1 }. Selidiki apakah fungsi f(x) kontiniu di x = -1.
Solution
We are given a piecewise function :
x^2 - 1, & x \leq -1 \\ 2x + 2, & x > -1 \end{cases}$$ We are tasked with investigating whether the function is continuous at $$x = -1$$. To check continuity at $$x = -1$$, we need to satisfy the following three conditions: 1. **Existence of $$f(-1)$$**: Since $$f(x) = x^2 - 1$$ when $$x \leq -1$$, we calculate: $$f(-1) = (-1)^2 - 1 = 1 - 1 = 0$$ So, $$f(-1) = 0$$. 2. **Existence of the left-hand limit** ($$\lim_{x \to -1^-} f(x)$$): As $$x$$ approaches -1 from the left ($$x \leq -1$$), the function is $$f(x) = x^2 - 1$$. $$\lim_{x \to -1^-} f(x) = (-1)^2 - 1 = 0$$ 3. **Existence of the right-hand limit** ($$\lim_{x \to -1^+} f(x)$$): As $$x$$ approaches -1 from the right ($$x > -1$$), the function is $$f(x) = 2x + 2$$. $$\lim_{x \to -1^+} f(x) = 2(-1) + 2 = -2 + 2 = 0$$ Since: - $$f(-1) = 0$$, - $$\lim_{x \to -1^-} f(x) = 0$$, - $$\lim_{x \to -1^+} f(x) = 0$$, and all these values are equal, the function $$f(x)$$ is continuous at $$x = -1$$. Would you like a more detailed explanation or have any questions? --- Here are 5 related questions to explore further: 1. What is the definition of a continuous function? 2. How do we define one-sided limits, and why are they important for continuity? 3. What are some common types of discontinuities (e.g., jump, removable)? 4. How can a piecewise function like this one be continuous everywhere? 5. How does the concept of differentiability relate to continuity? **Tip**: For checking continuity at any point, always make sure to verify both the function's value at the point and the limits from both sides!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Limits
Formulas
Limit from the left: lim(x→-1^-) f(x)
Limit from the right: lim(x→-1^+) f(x)
Continuity criterion: f(x) is continuous at x = c if lim(x→c^-) f(x) = lim(x→c^+) f(x) = f(c)
Theorems
Continuity Theorem
Suitable Grade Level
Grades 10-12