Determine the value of k for which the function f(x) = { x^2 + k, x < 1; 2x +

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Question: Determine the value of k for which the function f(x) = { x^2 + k, x < 1; 2x + 1, x >= 1 } is continuous at x = 1.

Options:

Correct Answer: 1

Solution:

To ensure continuity at x = 1, we need to set the two pieces equal: 1^2 + k = 2(1) + 1. This gives k = 2.

Determine the value of k for which the function f(x) = { x^2 + k, x < 1; 2x + 1, x >= 1 } is continuous at x = 1.

Determine the value of k for which the function f(x) = { x^2 + k, x < 1; 2x + 1, x >= 1 } is continuous at x = 1.

Correct Answer: 2

Step 1: Understand that the function f(x) has two parts: one for x < 1 and another for x >= 1.
Step 2: Identify the value of x where we want to check continuity, which is x = 1.
Step 3: Calculate the value of the first part of the function at x = 1. This is f(1) from the first part: f(1) = 1^2 + k.
Step 4: Calculate the value of the second part of the function at x = 1. This is f(1) from the second part: f(1) = 2(1) + 1.
Step 5: Set the two results from Step 3 and Step 4 equal to each other to ensure continuity: 1^2 + k = 2(1) + 1.
Step 6: Simplify the equation: 1 + k = 2 + 1.
Step 7: Solve for k: k = 2 + 1 - 1, which simplifies to k = 2.

Piecewise Functions – Understanding how to evaluate and ensure continuity in piecewise-defined functions.
Continuity at a Point – The requirement that the left-hand limit, right-hand limit, and function value at a point must be equal for continuity.
Solving for Variables – Setting equations equal to solve for unknowns in the context of continuity.