If f(x) = { x^2, x < 1; kx + 1, x >= 1 } is continuous at x = 1, find k.
Practice Questions
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If f(x) = { x^2, x < 1; kx + 1, x >= 1 } is continuous at x = 1, find k.
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Questions & Step-by-Step Solutions
If f(x) = { x^2, x < 1; kx + 1, x >= 1 } is continuous at x = 1, find k.
Correct Answer: 0
Step 1: Understand that the function f(x) has two parts: one for x less than 1 (f(x) = x^2) and one for x greater than or equal to 1 (f(x) = kx + 1).
Step 2: Since we want the function to be continuous at x = 1, the value of f(x) from the left side (when x is just less than 1) must equal the value of f(x) from the right side (when x is just greater than or equal to 1).
Step 3: Calculate f(1) using the left side: f(1) = 1^2 = 1.
Step 4: Calculate f(1) using the right side: f(1) = k(1) + 1 = k + 1.
Step 5: Set the two results equal to each other: 1 = k + 1.
Step 6: Solve for k by subtracting 1 from both sides: 1 - 1 = k + 1 - 1, which simplifies to 0 = k.
Step 7: Therefore, k = 0.
Piecewise Functions – Understanding how to evaluate and ensure continuity at a point for functions defined in pieces.
Continuity – The concept that a function is continuous at a point if the limit from both sides equals the function's value at that point.
Solving for Constants – Finding the value of a constant that ensures the function meets certain conditions, such as continuity.
