Math Problem Statement
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
k = For the value(s) of k,
f(x) ---Select--- is is not
continuous on
(−∞, ∞).
Solution
To find the value(s) of that make the function continuous at , we need to ensure that the left-hand limit, the right-hand limit, and the function value at are all equal.
The function is defined as:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Limit of ln(2x + 6) as x approaches -1
Limit of 7x - k as x approaches -1
Theorems
Continuity at a Point
Equal Left-Hand and Right-Hand Limits
Suitable Grade Level
Grades 11-12
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