Math Problem Statement
Suppose that the function f is defined, for all real numbers, as follows: f(x) = {x - 3 if x <= -2; 4x + 5 if x > -2}. Graph the function and determine whether or not it is continuous.
Solution
The problem you're dealing with asks whether the given function is continuous. Let's analyze the function step by step.
The function is defined piecewise as:
\begin{cases} x - 3 & \text{if } x \leq -2 \\ 4x + 5 & \text{if } x > -2 \end{cases}$$ ### Step 1: Continuity Check at $$x = -2$$ A function is continuous at a point if the following conditions hold: 1. The function is defined at that point. 2. The limit of the function as $$x$$ approaches the point from both sides exists. 3. The limit of the function as $$x$$ approaches the point from both sides is equal to the function's value at that point. #### Left-hand limit $$\lim_{x \to -2^-} f(x)$$: From the left-hand side ($$x \leq -2$$), the function is $$f(x) = x - 3$$. $$\lim_{x \to -2^-} f(x) = (-2) - 3 = -5$$ #### Right-hand limit $$\lim_{x \to -2^+} f(x)$$: From the right-hand side ($$x > -2$$), the function is $$f(x) = 4x + 5$$. $$\lim_{x \to -2^+} f(x) = 4(-2) + 5 = -8 + 5 = -3$$ ### Step 2: Compare the Limits - Left-hand limit: $$-5$$ - Right-hand limit: $$-3$$ Since the left-hand and right-hand limits are not equal, the function has a discontinuity at $$x = -2$$. ### Conclusion: The function is **not continuous** because the limits from the left and right at $$x = -2$$ do not match. Would you like more details on how this conclusion was reached? --- Here are five related questions you could explore: 1. How do you graph piecewise functions step by step? 2. What is the difference between removable and jump discontinuities? 3. How do you determine the domain of a piecewise function? 4. How do you compute limits at points of discontinuity? 5. Can a function be continuous but not differentiable? **Tip:** Always check both one-sided limits when determining the continuity at a boundary point of a piecewise function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Limits
Continuity
Formulas
Left-hand limit: lim(x -> -2^-) f(x) = (-2) - 3 = -5
Right-hand limit: lim(x -> -2^+) f(x) = 4(-2) + 5 = -3
Theorems
Limit Definition of Continuity
Suitable Grade Level
Grades 11-12
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