Math Problem Statement
Find AA given that the function, f(x)f(x), is continuous at 4.
f(x)={x2Ax−8x<4x≥4f(x)={x2x<4Ax−8x≥4
6
7
2
4
5
Solution
To find the value of such that the function is continuous at , let's consider the given function:
\begin{cases} x^2 & \text{if } x < 4 \\ Ax - 8 & \text{if } x \geq 4 \end{cases}$$ For $$f(x)$$ to be continuous at $$x = 4$$, the left-hand limit as $$x$$ approaches 4 from the left must equal the right-hand limit as $$x$$ approaches 4 from the right, and both must equal the function value at $$x = 4$$. ### Step 1: Compute the left-hand limit as $$x \to 4^-$$ For $$x < 4$$, $$f(x) = x^2$$. $$\lim_{x \to 4^-} f(x) = 4^2 = 16$$ ### Step 2: Compute the right-hand limit as $$x \to 4^+$$ For $$x \geq 4$$, $$f(x) = Ax - 8$$. $$\lim_{x \to 4^+} f(x) = A(4) - 8 = 4A - 8$$ ### Step 3: Set the limits equal to ensure continuity For continuity at $$x = 4$$, the limits from both sides must be equal: $$\lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x)$$ $$16 = 4A - 8$$ ### Step 4: Solve for $$A$$ Now, solve the equation $$16 = 4A - 8$$: $$16 + 8 = 4A$$ $$24 = 4A$$ $$A = \frac{24}{4} = 6$$ ### Conclusion The value of $$A$$ that makes the function $$f(x)$$ continuous at $$x = 4$$ is $$\boxed{6}$$. Would you like a further explanation on this? Here are some related questions: 1. What are the conditions for a function to be continuous at a point? 2. How do you handle continuity at points where the function changes its form? 3. Can a function be continuous but not differentiable at a point? 4. How do you find the limit of more complex piecewise functions? 5. What is the significance of ensuring continuity in a function? **Tip:** Always equate the left-hand limit and right-hand limit at the point of interest to ensure continuity in piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Continuity
Piecewise Functions
Formulas
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Theorems
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Suitable Grade Level
Advanced High School