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We are asked to determine for what values of \( a \), the given function \( f(x) \) is continuous at \( x = 3 \).

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

\[
f(x) = 
\begin{cases} 
a^2 + x^2 & \text{for } x < 3 \\
a(x + 3) & \text{for } x \geq 3
\end{cases}
\]

To ensure continuity at \( x = 3 \), the following condition must hold:

\[
\lim_{x \to 3^-} f(x) = f(3) = \lim_{x \to 3^+} f(x)
\]

### Step 1: Evaluate \( f(3) \)
Since \( f(x) = a(x + 3) \) for \( x \geq 3 \), we can directly substitute \( x = 3 \):

\[
f(3) = a(3 + 3) = a(6) = 6a
\]

### Step 2: Evaluate \( \lim_{x \to 3^-} f(x) \)
For \( x < 3 \), \( f(x) = a^2 + x^2 \). Taking the limit as \( x \to 3^- \):

\[
\lim_{x \to 3^-} f(x) = a^2 + 3^2 = a^2 + 9
\]

### Step 3: Evaluate \( \lim_{x \to 3^+} f(x) \)
For \( x \geq 3 \), \( f(x) = a(x + 3) \). Taking the limit as \( x \to 3^+ \):

\[
\lim_{x \to 3^+} f(x) = a(3 + 3) = 6a
\]

### Step 4: Set the limits equal for continuity
For the function to be continuous at \( x = 3 \), we need:

\[
a^2 + 9 = 6a
\]

### Step 5: Solve for \( a \)
Rearranging the equation:

\[
a^2 - 6a + 9 = 0
\]

This is a quadratic equation:

\[
(a - 3)^2 = 0
\]

Thus, \( a = 3 \).

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

Would you like more details or have any other questions?

Here are 5 related questions to extend your understanding:

1. What are the conditions for a piecewise function to be continuous at a point?
2. How does the choice of \( a \) affect the behavior of the function for \( x < 3 \)?
3. Can a function be continuous but not differentiable at a point? If so, under what circumstances?
4. How do you handle continuity conditions for piecewise functions with more than two pieces?
5. What are the differences between left-hand and right-hand limits?

**Tip:** When analyzing continuity, always check the left-hand and right-hand limits to ensure they match at the point in question!

Let f be the function defined as: f(x) = { a^2 + x^2 for x < 3, a(x + 3) for x ≥ 3 }, where a is a constant. For what values of a, if any, is f continuous at x = 3?

Explore how to determine the values of a constant 'a' that make the piecewise function f(x) continuous at x = 3. The problem involves evaluating limits and solving a quadratic equation, making it ideal for high school students.

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