https://file.aiquickdraw.com/mathbot/file/1730488964927y067o02t.jpg

The image shows a piecewise function defined as follows:

\[
f(x) = 
\begin{cases} 
x - 3 & \text{if } x \leq 2 \\
-5x + 4 & \text{if } x > 2 
\end{cases}
\]

### To analyze this function, let's look at a few properties:

1. **For \( x \leq 2 \):** The function follows the expression \( f(x) = x - 3 \).
   - For example, if \( x = 2 \), then \( f(2) = 2 - 3 = -1 \).

2. **For \( x > 2 \):** The function follows the expression \( f(x) = -5x + 4 \).
   - For example, if \( x = 3 \), then \( f(3) = -5(3) + 4 = -15 + 4 = -11 \).

### Analyzing Continuity at \( x = 2 \):

To determine if \( f(x) \) is continuous at \( x = 2 \), we need to check if the left-hand limit and the right-hand limit at \( x = 2 \) are equal to \( f(2) \).

1. **Left-hand limit as \( x \to 2^- \):**
   - Using \( f(x) = x - 3 \), the left-hand limit is \( 2 - 3 = -1 \).

2. **Right-hand limit as \( x \to 2^+ \):**
   - Using \( f(x) = -5x + 4 \), the right-hand limit is \( -5(2) + 4 = -10 + 4 = -6 \).

Since the left-hand limit (\(-1\)) and the right-hand limit (\(-6\)) are not equal, the function is **not continuous at \( x = 2 \)**.

Would you like further details or have any specific questions on this? Here are some related questions to explore:

1. What is the domain and range of \( f(x) \)?
2. How would you graph this piecewise function?
3. What are the limits of \( f(x) \) as \( x \to \infty \) and \( x \to -\infty \)?
4. Is there any interval where \( f(x) \) is increasing or decreasing?
5. How would you evaluate \( f(x) \) for specific values, such as \( x = 1 \) or \( x = 4 \)?

**Tip:** For piecewise functions, always check for continuity at the boundary points by comparing left-hand and right-hand limits.

Analyze the piecewise function: f(x) = { x - 3, if x ≤ 2; -5x + 4, if x > 2 }

Explore the properties of a piecewise function with expressions f(x) = x - 3 for x ≤ 2 and f(x) = -5x + 4 for x > 2. Learn how to analyze its continuity at x = 2 by evaluating left-hand and right-hand limits, suitable for students in Grades 10-12.

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