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The function \( f(x) \) given is a piecewise function defined as:

\[
f(x) = 
\begin{cases} 
-x + 2 & \text{if } x \leq 1 \\
5x - 2 & \text{if } x > 1
\end{cases}
\]

### Steps to Analyze the Continuity:

1. **Graphing the Function:**
   - For \( x \leq 1 \), the function is \( f(x) = -x + 2 \), which is a straight line with slope \(-1\) and y-intercept at 2.
   - For \( x > 1 \), the function is \( f(x) = 5x - 2 \), which is a straight line with slope 5 and y-intercept \(-2\).
   
   To graph this function:
   - Plot the line \( f(x) = -x + 2 \) for values of \( x \leq 1 \). This line passes through the points \( (0, 2) \) and \( (1, 1) \).
   - Plot the line \( f(x) = 5x - 2 \) for \( x > 1 \). This line passes through \( (1, 3) \) and continues with slope 5.

2. **Determining Continuity at \( x = 1 \):**
   To check if the function is continuous at \( x = 1 \), we need to verify:
   - \( \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1) \).

   - **Left-hand limit** (as \( x \to 1^- \)): 
     \[
     \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (-x + 2) = -(1) + 2 = 1.
     \]
   
   - **Right-hand limit** (as \( x \to 1^+ \)): 
     \[
     \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (5x - 2) = 5(1) - 2 = 3.
     \]
   
   - **Value of the function at \( x = 1 \)**:
     \[
     f(1) = -1 + 2 = 1 \quad (\text{since we use the first piece for } x \leq 1).
     \]

### Conclusion:
The function is **not continuous** at \( x = 1 \), because the left-hand limit \( 1 \), the right-hand limit \( 3 \), and the value of the function at \( x = 1 \) are not equal. There is a jump discontinuity at \( x = 1 \).

Would you like me to graph the function for you? Let me know if you need more details or clarifications!

### Questions:
1. What is the formal definition of continuity at a point?
2. Can you think of other types of discontinuities besides a jump discontinuity?
3. How do we determine if a piecewise function is differentiable at the joining point?
4. How does the slope affect the behavior of each segment of the function?
5. Can you find other points where the function might be discontinuous based on its definition?

### Tip:
For piecewise functions, always check the limits from both sides at the points where the function pieces meet to determine continuity or differentiability.

Graph the function f(x) = -x + 2 if x ≤ 1, and f(x) = 5x - 2 if x > 1. Then determine whether or not the function is continuous.

Explore how to graph the piecewise function f(x) defined by -x + 2 for x ≤ 1 and 5x - 2 for x > 1. Learn how to determine if the function is continuous at x = 1 by evaluating limits and the value of the function.

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