Math Problem Statement
solve the question given in the image.
Let f(x) = { ax + 4, x < -2; x^2 - 6, x ≥ -2 }. If f is continuous at x = -2, then find the value of a.
Solution
To solve the problem, we need to determine the value of so that the function is continuous at .
Given function:
\begin{cases} ax + 4 & \text{if } x < -2, \\ x^2 - 6 & \text{if } x \geq -2. \end{cases}$$ ### Condition for continuity: For the function to be continuous at $$x = -2$$, the following condition must hold: $$\lim_{x \to -2^-} f(x) = \lim_{x \to -2^+} f(x) = f(-2).$$ ### Step 1: Evaluate $$f(-2)$$ Since $$f(x) = x^2 - 6$$ for $$x \geq -2$$, we substitute $$x = -2$$ into this expression: $$f(-2) = (-2)^2 - 6 = 4 - 6 = -2.$$ ### Step 2: Left-hand limit $$\lim_{x \to -2^-} f(x)$$ For $$x < -2$$, $$f(x) = ax + 4$$. We compute the left-hand limit as $$x \to -2^-$$: $$\lim_{x \to -2^-} f(x) = \lim_{x \to -2^-} (ax + 4) = a(-2) + 4 = -2a + 4.$$ ### Step 3: Right-hand limit $$\lim_{x \to -2^+} f(x)$$ For $$x \geq -2$$, $$f(x) = x^2 - 6$$. We compute the right-hand limit as $$x \to -2^+$$: $$\lim_{x \to -2^+} f(x) = (-2)^2 - 6 = 4 - 6 = -2.$$ ### Step 4: Equating the limits for continuity To ensure continuity at $$x = -2$$, the left-hand limit must equal the right-hand limit: $$-2a + 4 = -2.$$ ### Step 5: Solve for $$a$$ $$-2a + 4 = -2 \implies -2a = -2 - 4 \implies -2a = -6 \implies a = 3.$$ ### Conclusion: The value of $$a$$ that makes the function continuous at $$x = -2$$ is $$\boxed{3}$$. Let me know if you'd like further clarification or a detailed explanation of any step! --- Here are 5 related questions: 1. What does it mean for a function to be continuous at a point? 2. How would the problem change if the condition for continuity was at a different value of $$x$$? 3. How do you determine the left-hand and right-hand limits for a piecewise function? 4. Can a function be discontinuous but still have a limit at a given point? 5. What are the general steps to find the value of a parameter in a continuity problem? **Tip**: In piecewise functions, always check the boundaries where the function's definition changes for continuity or differentiability conditions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity of functions
Limits
Piecewise functions
Formulas
lim_{x -> c^-} f(x) = lim_{x -> c^+} f(x) = f(c) for continuity at x = c
Theorems
Continuity theorem
Suitable Grade Level
Grades 11-12
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