Math Problem Statement
Let f(x) = { 8x - 5 if x <= 2, -5x + b if x > 2 }. If f(x) is a function which is continuous everywhere, then we must have b = ?
Solution
The piecewise function is given as:
\begin{cases} 8x - 5 & \text{if } x \leq 2 \\ -5x + b & \text{if } x > 2 \end{cases}$$ ### Step 1: Ensuring Continuity at $$x = 2$$ For $$f(x)$$ to be continuous everywhere, the left-hand limit (from the $$x \leq 2$$ part) and the right-hand limit (from the $$x > 2$$ part) must be equal at $$x = 2$$. #### Left-hand limit: $$\lim_{x \to 2^-} f(x) = 8(2) - 5 = 16 - 5 = 11$$ #### Right-hand limit: $$\lim_{x \to 2^+} f(x) = -5(2) + b = -10 + b$$ ### Step 2: Setting the limits equal Since the function must be continuous at $$x = 2$$, we equate the two limits: $$11 = -10 + b$$ ### Step 3: Solving for $$b$$ $$b = 11 + 10 = 21$$ Thus, the value of $$b$$ must be **21** for the function to be continuous everywhere. Would you like to see the graph of the function or explore any other details? Here are 5 related questions: 1. What does it mean for a function to be continuous at a point? 2. How would you check continuity at other points for different types of functions? 3. What are the key differences between piecewise functions and regular functions? 4. How do limits relate to continuity in functions? 5. Can you explain how derivatives apply to piecewise functions? **Tip:** When solving continuity problems, always ensure the limits from both sides of a critical point are equal.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity of functions
Piecewise functions
Limits
Formulas
Left-hand limit = Right-hand limit for continuity
Theorems
Continuity theorem at a point
Suitable Grade Level
Grades 10-12