Math Problem Statement
Find the value of the parameter \( a \) for which the function \( f(x) \) is continuous on the entire real line, where:
\[
f(x) = \begin{cases} x^2 + 1, & \text{if } x \leq 2, \\ x - a, & \text{if } x > 2. \end{cases}
\]
Solution
To solve this problem, we need to determine the value of the parameter that makes the function continuous on the entire real number line.
The function is defined as follows:
x^2 + 1, & \text{if } x \leq 2, \\ x - a, & \text{if } x > 2. \end{cases}$$ ### Step-by-Step Solution 1. **Continuity Condition**: For $$f(x)$$ to be continuous at $$x = 2$$, the left-hand limit (as $$x \to 2^-$$) and the right-hand limit (as $$x \to 2^+$$) must be equal, and both must equal $$f(2)$$. 2. **Left-Hand Limit (as $$x \to 2^-$$)**: $$\lim_{x \to 2^-} f(x) = 2^2 + 1 = 4 + 1 = 5.$$ 3. **Right-Hand Limit (as $$x \to 2^+$$)**: $$\lim_{x \to 2^+} f(x) = 2 - a.$$ 4. **Setting Limits Equal**: For continuity, the left-hand and right-hand limits must be equal: $$5 = 2 - a.$$ 5. **Solving for $$a$$**: $$a = 2 - 5 = -3.$$ ### Answer The value of $$a$$ that makes the function $$f(x)$$ continuous is: $$\boxed{-3}$$ Would you like more details on any part of this solution, or have any other questions? --- Here are five related questions that can expand your understanding: 1. What would happen to the continuity if we changed the function to $$x^2 + c$$ instead of $$x^2 + 1$$ for $$x \leq 2$$? 2. How can we determine if a function is differentiable at a given point of continuity? 3. What is the difference between continuity and differentiability for a function at a specific point? 4. How would the solution change if the break point was at $$x = 3$$ instead of $$x = 2$$? 5. Can a function be continuous but not differentiable? If so, could you provide an example? **Tip:** When working with piecewise functions, always check points where the definition of the function changes to ensure continuity by matching limits at those points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
Left-hand limit = Right-hand limit for continuity at a point
Theorems
Limit Definition of Continuity
Suitable Grade Level
High School
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