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To solve the problem, we need to determine the value of \( a \) so that the function \( f(x) \) is continuous at \( x = -2 \).

### Given function:
\[
f(x) = 
\begin{cases} 
ax + 4 & \text{if } x < -2, \\
x^2 - 6 & \text{if } x \geq -2.
\end{cases}
\]

### Condition for continuity:
For the function to be continuous at \( x = -2 \), the following condition must hold:
\[
\lim_{x \to -2^-} f(x) = \lim_{x \to -2^+} f(x) = f(-2).
\]

### Step 1: Evaluate \( f(-2) \)
Since \( f(x) = x^2 - 6 \) for \( x \geq -2 \), we substitute \( x = -2 \) into this expression:
\[
f(-2) = (-2)^2 - 6 = 4 - 6 = -2.
\]

### Step 2: Left-hand limit \( \lim_{x \to -2^-} f(x) \)
For \( x < -2 \), \( f(x) = ax + 4 \). We compute the left-hand limit as \( x \to -2^- \):
\[
\lim_{x \to -2^-} f(x) = \lim_{x \to -2^-} (ax + 4) = a(-2) + 4 = -2a + 4.
\]

### Step 3: Right-hand limit \( \lim_{x \to -2^+} f(x) \)
For \( x \geq -2 \), \( f(x) = x^2 - 6 \). We compute the right-hand limit as \( x \to -2^+ \):
\[
\lim_{x \to -2^+} f(x) = (-2)^2 - 6 = 4 - 6 = -2.
\]

### Step 4: Equating the limits for continuity
To ensure continuity at \( x = -2 \), the left-hand limit must equal the right-hand limit:
\[
-2a + 4 = -2.
\]

### Step 5: Solve for \( a \)
\[
-2a + 4 = -2 \implies -2a = -2 - 4 \implies -2a = -6 \implies a = 3.
\]

### Conclusion:
The value of \( a \) that makes the function continuous at \( x = -2 \) is \( \boxed{3} \).

Let me know if you'd like further clarification or a detailed explanation of any step!

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Here are 5 related questions:
1. What does it mean for a function to be continuous at a point?
2. How would the problem change if the condition for continuity was at a different value of \( x \)?
3. How do you determine the left-hand and right-hand limits for a piecewise function?
4. Can a function be discontinuous but still have a limit at a given point?
5. What are the general steps to find the value of a parameter in a continuity problem?

**Tip**: In piecewise functions, always check the boundaries where the function's definition changes for continuity or differentiability conditions.

Let f(x) = { ax + 4, x < -2; x^2 - 6, x ≥ -2 }. If f is continuous at x = -2, then find the value of a.

Learn how to solve a continuity problem for a piecewise function f(x) = { ax + 4, x < -2; x^2 - 6, x ≥ -2 }. This problem covers key concepts such as continuity, limits, and solving for a parameter. Suitable for students in Grades 11-12.

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