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We are given a piecewise function \( f(x) \):

\[
f(x) = \begin{cases} 
x^2 - 1, & x \leq -1 \\
2x + 2, & x > -1
\end{cases}
\]

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?

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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!

Diketahui f(x) = { x^2 - 1, x ≤ -1 ; 2x + 2, x > -1 }. Selidiki apakah fungsi f(x) kontiniu di x = -1.

This problem explores the continuity of a piecewise function at x = -1 by examining the limits from the left and right, as well as the function's value at x = -1. The solution involves applying the continuity theorem and checking whether the function satisfies the conditions for continuity.

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