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To find the value of \( k \) that makes the given piecewise function continuous, we need to ensure that the function is continuous at the point where the two pieces of the function meet, which is at \( x = -2 \).

The function is given by:

\[
f(x) = 
\begin{cases} 
kx^2 & \text{if } x \geq -2 \\
kx + 18 & \text{if } x < -2 
\end{cases}
\]

### Steps to ensure continuity at \( x = -2 \):
1. **For continuity**, the left-hand limit and the right-hand limit at \( x = -2 \) must be equal to the function's value at \( x = -2 \).
   
2. **Right-hand limit**: This is from the part of the function \( f(x) = kx^2 \) for \( x \geq -2 \). Substituting \( x = -2 \):
   \[
   f(-2) = k(-2)^2 = 4k
   \]

3. **Left-hand limit**: This is from the part of the function \( f(x) = kx + 18 \) for \( x < -2 \). Substituting \( x = -2 \):
   \[
   f(-2) = k(-2) + 18 = -2k + 18
   \]

4. **Set the right-hand limit equal to the left-hand limit** to ensure continuity:
   \[
   4k = -2k + 18
   \]

5. **Solve for \( k \)**:
   \[
   4k + 2k = 18
   \]
   \[
   6k = 18
   \]
   \[
   k = 3
   \]

Thus, the value of \( k \) that makes the function continuous is \( \boxed{3} \).

Would you like a more detailed explanation or have any other questions? 

Here are some related questions:
1. What happens to the graph of a function at points of discontinuity?
2. How do you determine if a piecewise function is differentiable?
3. Can a function be continuous but not differentiable at a point?
4. How do left-hand and right-hand limits help in proving continuity?
5. What is the difference between removable and non-removable discontinuities?

**Tip**: Always remember that continuity at a point means the function's value and its limits from both directions must match at that point.

Find the value of k that makes the function continuous: f(x) = {kx^2, x ≥ -2; kx + 18, x < -2}.

Discover how to determine the value of k that makes a piecewise function continuous. The function involves quadratic and linear expressions, with step-by-step explanations on setting right-hand and left-hand limits equal at x = -2. Suitable for high school students studying limits and continuity.

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