Math Problem Statement
Analyze the piecewise function: f(x) = { x - 3, if x ≤ 2; -5x + 4, if x > 2 }
Solution
The image shows a piecewise function defined as follows:
\begin{cases} x - 3 & \text{if } x \leq 2 \\ -5x + 4 & \text{if } x > 2 \end{cases}$$ ### To analyze this function, let's look at a few properties: 1. **For $$x \leq 2$$:** The function follows the expression $$f(x) = x - 3$$. - For example, if $$x = 2$$, then $$f(2) = 2 - 3 = -1$$. 2. **For $$x > 2$$:** The function follows the expression $$f(x) = -5x + 4$$. - For example, if $$x = 3$$, then $$f(3) = -5(3) + 4 = -15 + 4 = -11$$. ### Analyzing Continuity at $$x = 2$$: To determine if $$f(x)$$ is continuous at $$x = 2$$, we need to check if the left-hand limit and the right-hand limit at $$x = 2$$ are equal to $$f(2)$$. 1. **Left-hand limit as $$x \to 2^-$$:** - Using $$f(x) = x - 3$$, the left-hand limit is $$2 - 3 = -1$$. 2. **Right-hand limit as $$x \to 2^+$$:** - Using $$f(x) = -5x + 4$$, the right-hand limit is $$-5(2) + 4 = -10 + 4 = -6$$. Since the left-hand limit ($$-1$$) and the right-hand limit ($$-6$$) are not equal, the function is **not continuous at $$x = 2$$**. Would you like further details or have any specific questions on this? Here are some related questions to explore: 1. What is the domain and range of $$f(x)$$? 2. How would you graph this piecewise function? 3. What are the limits of $$f(x)$$ as $$x \to \infty$$ and $$x \to -\infty$$? 4. Is there any interval where $$f(x)$$ is increasing or decreasing? 5. How would you evaluate $$f(x)$$ for specific values, such as $$x = 1$$ or $$x = 4$$? **Tip:** For piecewise functions, always check for continuity at the boundary points by comparing left-hand and right-hand limits.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 at a Point
Limit Laws
Suitable Grade Level
Grades 10-12