Math Problem Statement
To what new value should f(2) be changed to remove the discontinuity? The piecewise function is defined as: f(x) = { 2x - 4, if x < 2; 2, if x = 2; x - 2, if x > 2 }.
Solution
To solve this problem, we need to ensure that the function is continuous at .
The piecewise function is defined as:
\begin{cases} 2x - 4 & \text{for } x < 2 \\ 2 & \text{for } x = 2 \\ x - 2 & \text{for } x > 2 \end{cases}$$ To remove the discontinuity at $$x = 2$$, the left-hand limit as $$x \to 2^-$$ (from the left) must equal the right-hand limit as $$x \to 2^+$$ (from the right), and both must equal $$f(2)$$. ### Step 1: Left-hand limit ($$x \to 2^-$$) For $$x < 2$$, the function is $$2x - 4$$. As $$x \to 2$$ from the left: $$\lim_{x \to 2^-} f(x) = 2(2) - 4 = 4 - 4 = 0$$ ### Step 2: Right-hand limit ($$x \to 2^+$$) For $$x > 2$$, the function is $$x - 2$$. As $$x \to 2$$ from the right: $$\lim_{x \to 2^+} f(x) = 2 - 2 = 0$$ ### Step 3: Adjust $$f(2)$$ To remove the discontinuity, $$f(2)$$ should be equal to the limit from both sides. Since both the left-hand and right-hand limits at $$x = 2$$ are 0, we should set $$f(2) = 0$$. Thus, the value of $$f(2)$$ should be changed to $$0$$ to remove the discontinuity. ### Answer: The correct choice is **D. 0**. Would you like more details or have any questions? --- Here are 5 related questions to expand on this topic: 1. What are the different types of discontinuities in a function? 2. How do you determine the continuity of a piecewise function in general? 3. Why is matching the left-hand and right-hand limits crucial for continuity? 4. How does the Intermediate Value Theorem relate to continuous functions? 5. Can a function be continuous at some points and discontinuous at others? **Tip:** Always check the limits from both sides at the point of interest when analyzing continuity in piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Formulas
Left-hand limit: lim(x→2-) f(x)
Right-hand limit: lim(x→2+) f(x)
Theorems
Limit Theorem
Definition of Continuity
Suitable Grade Level
Grades 9-11