## Continuity of a Piecewise Function### Problem StatementGiven the piecewise function below, find the value of $ k $ which makes the function continuous.\[ f(x) = \begin{cases} -x & \text{if } x < -3 \\k - 3x & \text{if } x \geq -3 \end{cases} \]### SolutionTo ensure the function $ f(x) $ is continuous at $ x = -3 $, the left-hand limit and the right-hand limit at $ x = -3 $ must be equal, and both must equal the value of the function at $ x = -3 $.#### Step-by-Step Process1. **Evaluate the left-hand limit** as $ x $ approaches $ -3 $: Since $ x < -3 $ for the left-hand limit, \[ \lim_{{x \to -3^-}} f(x) = \lim_{{x \to -3^-}} (-x) = -(-3) = 3. \]2. **Evaluate the right-hand limit** as $ x $ approaches $ -3 $: Since $ x \geq -3 $ for the right-hand limit, \[ \lim_{{x \to -3^+}} f(x) = \lim_{{x \to -3^+}} (k - 3x). \] Substitute $ x = -3 $ into $ k - 3x $: \[ f(-3) = k - 3(-3) = k + 9. \]3. **Set the left-hand limit equal to the right-hand limit** to ensure continuity: \[ 3 = k + 9. \] Solve for $ k $: \[ k = 3 - 9 = -6. \]### ConclusionThe value of $ k $ that makes the function continuous at $ x = -3 $ is $ \boxed{-6} $.

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Answered: Given the piecewise function below,…

Given the piecewise function below, find the value of which makes the function continuous. -I f(x): = if x < -3 k-3x if x ≥-3

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter2: Functions

Section: Chapter Questions

Problem 30P: In this problem you are asked to find a function that models in real life situation and then use the...

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Question

## Continuity of a Piecewise Function

### Problem Statement

Given the piecewise function below, find the value of $ k $ which makes the function continuous.

\[ f(x) = \begin{cases}
-x & \text{if } x < -3 \\
k - 3x & \text{if } x \geq -3
\end{cases} \]

### Solution

To ensure the function $ f(x) $ is continuous at $ x = -3 $, the left-hand limit and the right-hand limit at $ x = -3 $ must be equal, and both must equal the value of the function at $ x = -3 $.

#### Step-by-Step Process

1. **Evaluate the left-hand limit** as $ x $ approaches $ -3 $:

Since $ x < -3 $ for the left-hand limit,
\[
\lim_{{x \to -3^-}} f(x) = \lim_{{x \to -3^-}} (-x) = -(-3) = 3.
\]

2. **Evaluate the right-hand limit** as $ x $ approaches $ -3 $:

Since $ x \geq -3 $ for the right-hand limit,
\[
\lim_{{x \to -3^+}} f(x) = \lim_{{x \to -3^+}} (k - 3x).
\]
Substitute $ x = -3 $ into $ k - 3x $:
\[
f(-3) = k - 3(-3) = k + 9.
\]

3. **Set the left-hand limit equal to the right-hand limit** to ensure continuity:
\[
3 = k + 9.
\]
Solve for $ k $:
\[
k = 3 - 9 = -6.
\]

### Conclusion

The value of $ k $ that makes the function continuous at $ x = -3 $ is $ \boxed{-6} $.

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