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We are asked to determine whether the given function \( f(x) \) is continuous at \( x = 1 \).

The function is defined piecewise as:

\[
f(x) = 
\begin{cases} 
4x - 1 & \text{if } x \geq 1 \\
-x^2 + x + 3 & \text{if } x < 1 
\end{cases}
\]

To determine if \( f(x) \) is continuous at \( x = 1 \), we need to check the following conditions:

1. **Existence of \( f(1) \)**: 
   The function is defined by the expression \( 4x - 1 \) when \( x \geq 1 \). So,
   \[
   f(1) = 4(1) - 1 = 3
   \]

2. **Existence of the left-hand limit** \( \lim_{x \to 1^-} f(x) \): 
   For \( x < 1 \), the function is given by \( f(x) = -x^2 + x + 3 \). Thus, we calculate the limit as \( x \) approaches 1 from the left:
   \[
   \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (-x^2 + x + 3) = -(1)^2 + 1 + 3 = 3
   \]

3. **Existence of the right-hand limit** \( \lim_{x \to 1^+} f(x) \):
   For \( x \geq 1 \), the function is given by \( f(x) = 4x - 1 \). Thus, we calculate the limit as \( x \) approaches 1 from the right:
   \[
   \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (4x - 1) = 4(1) - 1 = 3
   \]

4. **Conclusion**:
   Since all the conditions are satisfied:
   - \( f(1) = 3 \)
   - \( \lim_{x \to 1^-} f(x) = 3 \)
   - \( \lim_{x \to 1^+} f(x) = 3 \)

   We conclude that the function \( f(x) \) is continuous at \( x = 1 \).

Would you like more details or have any questions? Here are some related questions for further exploration:

1. What are the general conditions for a function to be continuous at a point?
2. How would you define continuity in terms of limits for a piecewise function?
3. How does the behavior of a function change at a discontinuity?
4. Can a function be differentiable at a point where it is not continuous?
5. How would you approach checking the continuity of a function at multiple points?

**Tip**: Always check both the function's value at a point and the limits from both sides when determining continuity at that point.

The function f is defined by f(x) = {4x - 1 if x ≥ 1, -x^2 + x + 3 if x < 1}. Determine whether or not f is continuous at x = 1.

Explore how to determine whether the piecewise function f(x), defined by different expressions for x ≥ 1 and x < 1, is continuous at x = 1. This problem covers key concepts of limits, continuity, and piecewise functions, suitable for Grades 11-12.

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