To two decimal places, find the value of k that will make the function f(x) continuous everywhere.
f of x equals the quantity 3 times x plus k for x less than or equal to negative 4 and is equal to k times x squared minus 5 for x greater than negative 4 (4 points)


11.00

-2.47

-0.47

None of these

To make the function \( f(x) \) continuous everywhere, we need the two pieces of the function to meet at \( x = -4 \), meaning the left-hand limit and right-hand limit at \( x = -4 \) must be equal.

The given function is:

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

### Step 1: Find the value of the function for \( x = -4 \) for both pieces.

- Left-hand limit at \( x = -4 \) (from \( 3x + k \)):

\[
\lim_{x \to -4^-} f(x) = 3(-4) + k = -12 + k
\]

- Right-hand limit at \( x = -4 \) (from \( kx^2 - 5 \)):

\[
\lim_{x \to -4^+} f(x) = k(-4)^2 - 5 = k(16) - 5 = 16k - 5
\]

### Step 2: Set the two limits equal to each other for continuity:

\[
-12 + k = 16k - 5
\]

### Step 3: Solve for \( k \):

\[
-12 + k = 16k - 5
\]
\[
-12 + 5 = 16k - k
\]
\[
-7 = 15k
\]
\[
k = \frac{-7}{15} \approx -0.47
\]

Thus, the value of \( k \) that makes the function continuous is approximately **-0.47**.

### Final Answer:
**-0.47**

Let me know if you'd like further details or if you have any questions!

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**Further Questions:**
1. What is the concept of continuity in a piecewise function?
2. How do you solve equations involving limits in piecewise functions?
3. How do you find limits for other types of functions (e.g., rational or trigonometric)?
4. What are some common techniques for evaluating limits?
5. What happens if the two limits do not equal each other at a specific point?

**Tip:** Always check the behavior of both pieces of a piecewise function at the boundary point to ensure continuity.

To two decimal places, find the value of k that will make the function f(x) continuous everywhere. f of x equals the quantity 3 times x plus k for x less than or equal to negative 4 and is equal to k times x squared minus 5 for x greater than negative 4.

Learn how to find the value of k that makes a piecewise function continuous everywhere. This problem involves limits and continuity at the boundary of two function pieces. Solution involves algebraic manipulation to determine the correct value of k. Suitable for students in Grades 11-12.

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