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The problem you're dealing with asks whether the given function is continuous. Let's analyze the function step by step.

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

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

### Step 1: Continuity Check at \( x = -2 \)

A function is continuous at a point if the following conditions hold:
1. The function is defined at that point.
2. The limit of the function as \( x \) approaches the point from both sides exists.
3. The limit of the function as \( x \) approaches the point from both sides is equal to the function's value at that point.

#### Left-hand limit \( \lim_{x \to -2^-} f(x) \):
From the left-hand side (\( x \leq -2 \)), the function is \( f(x) = x - 3 \).
\[
\lim_{x \to -2^-} f(x) = (-2) - 3 = -5
\]

#### Right-hand limit \( \lim_{x \to -2^+} f(x) \):
From the right-hand side (\( x > -2 \)), the function is \( f(x) = 4x + 5 \).
\[
\lim_{x \to -2^+} f(x) = 4(-2) + 5 = -8 + 5 = -3
\]

### Step 2: Compare the Limits

- Left-hand limit: \( -5 \)
- Right-hand limit: \( -3 \)

Since the left-hand and right-hand limits are not equal, the function has a discontinuity at \( x = -2 \).

### Conclusion:
The function is **not continuous** because the limits from the left and right at \( x = -2 \) do not match.

Would you like more details on how this conclusion was reached?

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Here are five related questions you could explore:
1. How do you graph piecewise functions step by step?
2. What is the difference between removable and jump discontinuities?
3. How do you determine the domain of a piecewise function?
4. How do you compute limits at points of discontinuity?
5. Can a function be continuous but not differentiable?

**Tip:** Always check both one-sided limits when determining the continuity at a boundary point of a piecewise function.

Suppose that the function f is defined, for all real numbers, as follows: f(x) = {x - 3 if x <= -2; 4x + 5 if x > -2}. Graph the function and determine whether or not it is continuous.

Analyze the continuity of a piecewise function defined as f(x) = {x - 3 for x <= -2; 4x + 5 for x > -2}. Find the left-hand and right-hand limits and conclude whether the function is continuous at x = -2. Suitable for Grades 11-12.

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