Math Problem Statement
Let 𝑓f be the function
𝑓(𝑥)={𝑥2−𝑐4𝑥+2𝑐for 𝑥<5for 𝑥≥5 f(x)={x2−cfor x<54x+2cfor x≥5
Find the value of 𝑐c that makes the function continuous.
(Use symbolic notation and fractions where needed.)
Solution
To find the value of that makes the function continuous, we need to ensure that the left-hand limit (as approaches 5 from the left) equals the right-hand limit (as approaches 5 from the right), and that both equal the function value at .
The function is defined as:
x^2 - c & \text{for } x < 5 \\ 4x + 2c & \text{for } x \geq 5 \end{cases}$$ ### Step 1: Evaluate the limit as $$x$$ approaches 5 from the left. For $$x < 5$$, $$f(x) = x^2 - c$$. $$\lim_{x \to 5^-} f(x) = 5^2 - c = 25 - c$$ ### Step 2: Evaluate the limit as $$x$$ approaches 5 from the right. For $$x \geq 5$$, $$f(x) = 4x + 2c$$. $$\lim_{x \to 5^+} f(x) = 4(5) + 2c = 20 + 2c$$ ### Step 3: Set the limits equal to each other for continuity. To make $$f(x)$$ continuous at $$x = 5$$, the left-hand limit must equal the right-hand limit: $$25 - c = 20 + 2c$$ ### Step 4: Solve for $$c$$. $$25 - 20 = 2c + c$$ $$5 = 3c$$ $$c = \frac{5}{3}$$ Thus, the value of $$c$$ that makes the function $$f(x)$$ continuous is $$\boxed{\frac{5}{3}}$$. Would you like more details or have any questions? Here are some follow-up questions you might find interesting: 1. What does it mean for a function to be continuous at a point? 2. Can you describe the behavior of $$f(x)$$ at $$x = 5$$ when $$c = \frac{5}{3}$$? 3. What happens if $$c$$ takes a different value than $$\frac{5}{3}$$? 4. How would you find the value of $$c$$ if the function was piecewise-defined over a different interval? 5. Can a function be continuous but not differentiable at a certain point? **Tip:** Remember, for continuity at a point, the function must be defined at that point, and the left-hand and right-hand limits must both exist and be equal.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Limits
Continuity
Formulas
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Theorems
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Suitable Grade Level
Grades 11-12