Math Problem Statement
Solution
The image contains a problem asking for the value of the constant that makes a piecewise function continuous over the interval .
Given Function:
\begin{cases} x^2 - c & \text{for } x < 1, \\ cx + 5 & \text{for } x \geq 1 \end{cases}$$ **Objective:** Find the value of $$c$$ such that the function $$f(x)$$ is continuous on the entire real line, $$(-∞, ∞)$$. ### Steps to Solve: 1. **Continuity Condition at $$x = 1$$:** For $$f(x)$$ to be continuous at $$x = 1$$, the left-hand limit (as $$x$$ approaches 1 from the left) must equal the right-hand limit (as $$x$$ approaches 1 from the right), and both must equal $$f(1)$$. This can be expressed as: $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1)$$ 2. **Calculate the Left-Hand Limit:** $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1} (x^2 - c) = 1^2 - c = 1 - c$$ 3. **Calculate the Right-Hand Limit:** $$\lim_{x \to 1^+} f(x) = \lim_{x \to 1} (cx + 5) = c \cdot 1 + 5 = c + 5$$ 4. **Set the Left-Hand Limit Equal to the Right-Hand Limit:** $$1 - c = c + 5$$ 5. **Solve for $$c$$:** Combine like terms: $$1 - c = c + 5$$ $$1 - 5 = c + c$$ $$-4 = 2c$$ $$c = -2$$ ### Conclusion: The value of the constant $$c$$ that makes the function $$f(x)$$ continuous on the interval $$(-∞, ∞)$$ is $$\boxed{-2}$$. Would you like more details or have any questions about this solution? Here are 5 related questions to expand on this information: 1. How do you determine continuity for piecewise functions with more than two pieces? 2. What does it mean for a function to be differentiable at a point, and how does it relate to continuity? 3. Can a function be continuous at a point but not differentiable there? If so, provide an example. 4. How do we handle discontinuities in functions that are defined in different forms? 5. What are some common types of discontinuities in mathematical functions? **Tip:** For piecewise functions, always check the continuity at the points where the definition changes to ensure smooth transitions between pieces.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Formulas
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Theorems
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Suitable Grade Level
Grades 11-12