Math Problem Statement
Consider the following piecewise function:
f(x) =
{ 3x, if x < 0
2x^2, if 0 < x ≤ 5
2x + 3, if 5 < x }
Which of the following is true about this function?
Solution
Let's analyze the piecewise function:
\begin{cases} 3x, & \text{if } x < 0 \\ 2x^2, & \text{if } 0 \leq x \leq 5 \\ 2x + 3, & \text{if } x > 5 \end{cases}$$ We are asked to determine which of the following is true regarding the continuity and discontinuities of the function. Let's analyze each part: ### 1. **At $$x = 0$$:** - Left-hand limit ($$x \to 0^-$$): Using the first piece $$3x$$, $$\lim_{x \to 0^-} f(x) = 3(0) = 0$$ - Right-hand limit ($$x \to 0^+$$): Using the second piece $$2x^2$$, $$\lim_{x \to 0^+} f(x) = 2(0)^2 = 0$$ - Value of $$f(0)$$: Since $$f(x) = 2x^2$$ when $$x = 0$$, $$f(0) = 2(0)^2 = 0$$ Since the left-hand limit, right-hand limit, and the function value at $$x = 0$$ all match, **the function is continuous at $$x = 0$$**. ### 2. **At $$x = 5$$:** - Left-hand limit ($$x \to 5^-$$): Using the second piece $$2x^2$$, $$\lim_{x \to 5^-} f(x) = 2(5)^2 = 50$$ - Right-hand limit ($$x \to 5^+$$): Using the third piece $$2x + 3$$, $$\lim_{x \to 5^+} f(x) = 2(5) + 3 = 13$$ Since the left-hand limit ($$50$$) and the right-hand limit ($$13$$) are not equal, **there is a jump discontinuity at $$x = 5$$**. ### Conclusion: - The function is continuous at $$x = 0$$. - The function has a jump discontinuity at $$x = 5$$. ### Correct Answer: The first option: - "This function is continuous at $$x = 0$$, it has a jump discontinuity at $$x = 5$$." Would you like further clarification or more details? Here are 5 related questions to expand on this topic: 1. How do you identify whether a function has a jump discontinuity? 2. What are the different types of discontinuities in functions? 3. What is the importance of left-hand and right-hand limits in checking continuity? 4. Can a function have both removable and jump discontinuities at the same point? 5. How does the definition of a piecewise function affect its continuity properties? **Tip:** Always check both the limits and the actual value of the function at the points of interest when determining continuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Discontinuities
Formulas
Left-hand limit
Right-hand limit
Checking continuity by limits
Theorems
Continuity Theorem
Jump Discontinuity
Removable Discontinuity
Suitable Grade Level
Grade 12/College Level