Math Problem Statement
The value of k that will make the function f continuous on the reals. f(x) = {x + 6, if x <= 2; kx^2, if x > 2}. Options: k = 2, k = 0, k = 3, k = 6.
Solution
To ensure the function is continuous at , we need the values of both pieces of the function to be equal when . Given:
\begin{cases} x + 6, & \text{if } x \leq 2 \\ kx^2, & \text{if } x > 2 \end{cases}$$ 1. Evaluate the first piece at $$x = 2$$: $$f(2) = 2 + 6 = 8.$$ 2. For continuity, we must have: $$k \cdot 2^2 = 8.$$ 3. Solving for $$k$$: $$4k = 8 \implies k = 2.$$ Therefore, the value of $$k$$ that makes the function continuous is $$k = 2$$. Would you like more details on any part of this solution? Here are some related questions you might find helpful: 1. What does it mean for a function to be continuous at a point? 2. How do we check the continuity of piecewise functions? 3. Why is it necessary for both parts of a piecewise function to meet at a boundary point for continuity? 4. How would the solution change if the condition was $$x < 2$$ instead of $$x \leq 2$$? 5. What happens if we select an incorrect value of $$k$$ in terms of the graph of the function? **Tip:** Continuity at a point means the left-hand limit, right-hand limit, and the value of the function at that point must all be equal.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Limit of f(x) as x approaches a point from left equals limit from right for continuity
Theorems
Definition of Continuity at a Point
Suitable Grade Level
High School (Grades 10-12)