https://file.aiquickdraw.com/mathbot/file/1726027404252e1zg9myo.jpg

Let's break down the problem step-by-step:

### Given function:

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

### Steps to solve:

1. **Graphing the function:**
   - For \(x \leq -2\), the function is linear with the equation \(f(x) = x - 3\). This is a straight line with slope 1 and y-intercept \(-3\).
   - For \(x > -2\), the function is also linear, defined as \(f(x) = 4x + 5\). This is a straight line with slope 4 and y-intercept 5.

2. **Determining Continuity at \(x = -2\):**
   To determine if the function is continuous at \(x = -2\), we need to check the following conditions:
   - **Left-hand limit**: The limit of \(f(x)\) as \(x\) approaches \(-2\) from the left.
   - **Right-hand limit**: The limit of \(f(x)\) as \(x\) approaches \(-2\) from the right.
   - **Value of the function at \(x = -2\)**: \(f(-2)\).

### Calculation:

1. **Left-hand limit:**
   For \(x \leq -2\), the function is \(f(x) = x - 3\). So, as \(x \to -2^-\), we substitute \(x = -2\) into the expression:
   \[
   \lim_{x \to -2^-} f(x) = -2 - 3 = -5.
   \]

2. **Right-hand limit:**
   For \(x > -2\), the function is \(f(x) = 4x + 5\). As \(x \to -2^+\), we substitute \(x = -2\) into this expression:
   \[
   \lim_{x \to -2^+} f(x) = 4(-2) + 5 = -8 + 5 = -3.
   \]

3. **Value at \(x = -2\):**
   Since the function at \(x = -2\) is defined by the first piece \(f(x) = x - 3\) for \(x \leq -2\), we have:
   \[
   f(-2) = -2 - 3 = -5.
   \]

### Conclusion:
- The left-hand limit at \(x = -2\) is \(-5\).
- The right-hand limit at \(x = -2\) is \(-3\).
- The value of the function at \(x = -2\) is \(-5\).

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

Would you like me to graph the function for better visualization?

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Here are 5 related questions to consider:
1. What is the definition of continuity at a point?
2. How do you compute limits for piecewise functions?
3. What does it mean for a function to be differentiable at a point?
4. How does the slope of a line affect its behavior as \(x\) approaches a point?
5. What is the relationship between limits and continuity in calculus?

**Tip:** When working with piecewise functions, always check both the limits and the value of the function at the boundary points to determine continuity.

Graph the function f(x) = {x-3 if x <= -2, 4x+5 if x > -2}, and determine whether the function is continuous.

Graph the piecewise function f(x) and determine its continuity at x = -2. Explore the concepts of limits and continuity using a detailed step-by-step solution, including left-hand and right-hand limit evaluations.

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