Math Problem Statement
Find the value of c that makes f ( x ) continuous at x
2 : f ( x )
{ x 2 + c x ≤ 2 2 x − 1 x
2
Solution
To find the value of that makes the function continuous at , we need to ensure that the left-hand limit and the right-hand limit of as approaches 2 are equal, and that they are equal to the value of the function at .
Given Function
\begin{cases} x^2 + c & \text{if } x \leq 2 \\ 2x - 1 & \text{if } x > 2 \end{cases}$$ ### Steps to Solve 1. **Find the value of $$f(x)$$ at $$x = 2$$ using the first piece of the function:** $$f(2) = 2^2 + c = 4 + c$$ 2. **Find the limit of $$f(x)$$ as $$x$$ approaches 2 from the left (using $$x^2 + c$$):** $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x^2 + c) = 2^2 + c = 4 + c$$ 3. **Find the limit of $$f(x)$$ as $$x$$ approaches 2 from the right (using $$2x - 1$$):** $$\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (2x - 1) = 2 \cdot 2 - 1 = 4 - 1 = 3$$ 4. **Set the left-hand limit equal to the right-hand limit for continuity at $$x = 2$$:** $$4 + c = 3$$ 5. **Solve for $$c$$:** $$c = 3 - 4 = -1$$ ### Conclusion The value of $$c$$ that makes the function $$f(x)$$ continuous at $$x = 2$$ is $$\boxed{-1}.$$ Would you like more details on any part of the solution? Here are a few related questions you might consider: 1. How would the solution change if the function were discontinuous at a different point? 2. Can we apply this method to functions with more than two pieces? 3. What are the different types of discontinuities in a function? 4. How can we determine if a function is differentiable at a point? 5. What is the significance of the Intermediate Value Theorem in continuous functions? **Tip:** To check if a function is continuous at a specific point, always ensure that the left-hand limit, right-hand limit, and the value of the function at that point are all equal.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Formulas
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Theorems
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Suitable Grade Level
Grades 11-12