Math Problem Statement
Find the value of π so that π(π₯) is continuous everywhere
Solution
To ensure that the function is continuous everywhere, the function must be continuous at the point where the two pieces of the piecewise function meet, which is at .
The given function is:
\begin{cases} x^2 - 2x + 3, & \text{if } x \leq -2 \\ kx + 1, & \text{if } x > -2 \end{cases}$$ For the function to be continuous at $$x = -2$$, the two pieces must meet smoothly at $$x = -2$$. This means the limit of the function from the left ($$x \to -2^-$$) must equal the limit from the right ($$x \to -2^+$$) and both must equal the function value at $$x = -2$$. ### Step 1: Calculate $$g(-2)$$ from the first part of the piecewise function The left-hand side of the function is defined as $$g(x) = x^2 - 2x + 3$$ for $$x \leq -2$$. Substituting $$x = -2$$: $$g(-2) = (-2)^2 - 2(-2) + 3 = 4 + 4 + 3 = 11$$ Thus, $$g(-2) = 11$$. ### Step 2: Find the right-hand limit as $$x \to -2^+$$ For $$x > -2$$, the function is $$g(x) = kx + 1$$. Taking the limit as $$x$$ approaches $$-2$$ from the right: $$\lim_{x \to -2^+} g(x) = k(-2) + 1 = -2k + 1$$ ### Step 3: Set the two limits equal to ensure continuity For continuity at $$x = -2$$, the function values must be equal from both sides: $$g(-2) = \lim_{x \to -2^+} g(x)$$ $$11 = -2k + 1$$ ### Step 4: Solve for $$k$$ $$11 = -2k + 1$$ $$11 - 1 = -2k$$ $$10 = -2k$$ $$k = -5$$ ### Conclusion: The value of $$k$$ that makes the function $$g(x)$$ continuous everywhere is $$\boxed{-5}$$. Would you like more details or have any further questions? Here are 5 follow-up questions related to this problem: 1. How do we check if a function is differentiable at a point? 2. What happens if $$k$$ had been a different value? 3. How would the graph of this piecewise function look like with $$k = -5$$? 4. Can you explain the concept of continuity in a formal way? 5. What are the consequences if a function is not continuous at a point? **Tip:** Continuity ensures that there are no jumps or breaks in a function at a particular point, which is crucial in calculus, especially for differentiation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
g(x) = x^2 - 2x + 3, x β€ -2
g(x) = kx + 1, x > -2
Theorems
Continuity at a point
Limit definition of continuity
Suitable Grade Level
Grade 11-12 (Pre-Calculus/Calculus)
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