Math Problem Statement
Suppose that the function f is defined, for all real numbers, as follows:
f(x) =
{ x - 3 if x ≤ 2
{ -5x + 4 if x > 2
Graph the function f. Then determine whether or not the function is continuous.
Solution
To solve this problem, let’s analyze and graph the piecewise function given by:
\begin{cases} x - 3 & \text{if } x \leq 2 \\ -5x + 4 & \text{if } x > 2 \end{cases}$$ ### Step 1: Graph Each Piece 1. **For $$x \leq 2$$:** - The function is $$f(x) = x - 3$$, which is a linear function with slope 1 and y-intercept at $$-3$$. - Plugging in $$x = 2$$, we get $$f(2) = 2 - 3 = -1$$. - Plot this line for all $$x \leq 2$$, marking a solid point at $$(2, -1)$$ since $$x = 2$$ is included in this part. 2. **For $$x > 2$$:** - The function is $$f(x) = -5x + 4$$, another linear function with slope $$-5$$ and y-intercept at $$4$$. - Plugging in $$x = 2$$ (for the transition point), we get $$f(2) = -5(2) + 4 = -10 + 4 = -6$$. - Plot this line starting from an open point at $$(2, -6)$$, indicating that $$x = 2$$ is not included. ### Step 2: Determine Continuity A function is continuous at $$x = 2$$ if: 1. $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = f(2)$$. 2. For the left-hand limit as $$x \to 2^-$$, using $$f(x) = x - 3$$, we get $$\lim_{x \to 2^-} f(x) = -1$$. 3. For the right-hand limit as $$x \to 2^+$$, using $$f(x) = -5x + 4$$, we get $$\lim_{x \to 2^+} f(x) = -6$$. Since $$-1 \neq -6$$, the limits from both sides do not match, meaning the function is **not continuous at $$x = 2$$**. ### Answer: The function $$f(x)$$ is **not continuous** at $$x = 2$$. Would you like more details on this process, or have any questions? --- Here are some additional questions related to this topic: 1. What is the difference between left-hand and right-hand limits in piecewise functions? 2. How can we determine if a piecewise function is differentiable at a given point? 3. What does it mean for a function to have a removable discontinuity? 4. How would the graph look if the two parts of the function matched at $$x = 2$$? 5. How can piecewise functions model real-world situations where conditions change? **Tip:** When working with piecewise functions, always check continuity at the points where the function's definition changes.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Limits
Formulas
f(x) = x - 3 for x ≤ 2
f(x) = -5x + 4 for x > 2
Theorems
Continuity Theorem
Left-hand and Right-hand Limits
Suitable Grade Level
Grades 10-12