Find the value(s) of *k* such that the function is continuous at 

*x* = −1.

 (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

*f*(*x*) = 

ln(2*x* + 6),

     *x* < −1

7*x* − *k*,

     *x* ≥ −1

*k* = 
 
For the value(s) of *k*, 

*f*(*x*)  ---Select--- is is not

 continuous on 

(−∞, ∞).

To find the value(s) of \( k \) that make the function \( f(x) \) continuous at \( x = -1 \), we need to ensure that the left-hand limit, the right-hand limit, and the function value at \( x = -1 \) are all equal.

The function \( f(x) \) is defined as:
\[
f(x) =
\begin{cases}
\ln(2x + 6), & \text{if } x < -1 \\
7x - k, & \text{if } x \geq -1
\end{cases}
\]

### Step 1: Evaluate the left-hand limit as \( x \) approaches \(-1\):
For \( x < -1 \), the function is given by \( \ln(2x + 6) \). We compute the limit as \( x \) approaches \(-1\) from the left:
\[
\lim_{x \to -1^-} \ln(2x + 6) = \ln(2(-1) + 6) = \ln(4)
\]

### Step 2: Evaluate the right-hand limit as \( x \) approaches \(-1\):
For \( x \geq -1 \), the function is given by \( 7x - k \). We compute the limit as \( x \) approaches \(-1\) from the right:
\[
\lim_{x \to -1^+} (7x - k) = 7(-1) - k = -7 - k
\]

### Step 3: Set the left-hand limit equal to the right-hand limit for continuity:
For the function to be continuous at \( x = -1 \), we need:
\[
\ln(4) = -7 - k
\]

### Step 4: Solve for \( k \):
\[
k = -7 - \ln(4)
\]

Therefore, the value of \( k \) that makes the function continuous at \( x = -1 \) is:
\[
k = -7 - \ln(4)
\]

### Continuity on \( (-\infty, \infty) \):
Since the function is continuous at \( x = -1 \) when \( k = -7 - \ln(4) \), and the logarithmic function and linear function are continuous on their respective domains, \( f(x) \) is continuous on \( (-\infty, \infty) \) for this value of \( k \).

So, the final answers are:

1. \( k = -7 - \ln(4) \)
2. \( f(x) \) is continuous on \( (-\infty, \infty) \) for this value of \( k \).

Do you have any questions, or would you like further details?

Here are five questions that build on this information:
1. How do you determine the domain of a logarithmic function?
2. What is the significance of matching left-hand and right-hand limits for continuity?
3. Can a function be continuous if the left-hand limit, right-hand limit, and function value are not all equal at a given point?
4. How would the problem change if the inequality signs in the function definition were reversed?
5. What is the geometric interpretation of continuity at a point?

**Tip:** Always check both the left-hand and right-hand limits when determining continuity at a specific point.

Find the value(s) of k such that the function is continuous at x = −1.
(Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
f(x) = ln(2x + 6), x < −1
7x − k, x ≥ −1

Determine the value of k that ensures the continuity of a piecewise function at x = -1. This problem involves calculating limits and understanding continuity in piecewise functions, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation