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To determine the value of \( a \) such that the function \( f(x) \) is continuous at \( x = 3 \), we need to ensure that the left-hand limit, the right-hand limit, and the function's value at \( x = 3 \) all match.

### Given function:
\[
f(x) = 
\begin{cases} 
a^2 + x^2 & \text{for } x < 3, \\
a(x + 3) & \text{for } x \geq 3.
\end{cases}
\]

### Continuity condition at \( x = 3 \):
For continuity at \( x = 3 \), the left-hand limit as \( x \to 3^- \), the right-hand limit as \( x \to 3^+ \), and \( f(3) \) must all be equal.

1. **Left-hand limit \( \lim_{x \to 3^-} f(x) \)**:
   From the definition of \( f(x) \) when \( x < 3 \):
   \[
   \lim_{x \to 3^-} f(x) = a^2 + 3^2 = a^2 + 9.
   \]

2. **Right-hand limit \( \lim_{x \to 3^+} f(x) \)**:
   From the definition of \( f(x) \) when \( x \geq 3 \):
   \[
   \lim_{x \to 3^+} f(x) = f(3) = a(3 + 3) = 6a.
   \]

For \( f(x) \) to be continuous at \( x = 3 \), we must have:
\[
a^2 + 9 = 6a.
\]

### Solve the equation:
\[
a^2 - 6a + 9 = 0.
\]
This is a quadratic equation:
\[
(a - 3)^2 = 0.
\]
Thus, \( a = 3 \).

### Conclusion:
The function is continuous at \( x = 3 \) if and only if \( a = 3 \).

So, the correct answer is \( \boxed{B} \).

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Do you need further details or have any questions?

Here are 5 related questions to explore:
1. What is the definition of continuity at a point?
2. How do you find the limit of a piecewise function?
3. What happens if \( a \) takes other values in this function? 
4. Can a function be discontinuous at multiple points and still be differentiable?
5. How would the function's continuity change if the piecewise conditions were defined for different ranges?

**Tip:** Continuity at a point requires the function value and both one-sided limits to be equal. Always check for this condition when evaluating continuity.

For what values of a, if any, is the function f(x) continuous at x = 3?

Learn how to find the values of 'a' that make a piecewise function continuous at a specific point. This problem covers key mathematical concepts such as continuity, limits, and solving quadratic equations. Suitable for high school students in Grades 10-12.

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