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 + 3, x >= 1 } is continuous at x = 1.

Options:

Correct Answer: 0

Solution:

To ensure continuity at x = 1, we need to set the two pieces equal: k + 1^2 = 2(1) + 3. This gives k + 1 = 5, so k = 4.

Determine the value of k for which the function f(x) = { x^2 + k, x < 1; 2x + 3, x >= 1 } is continuous at x = 1.

Determine the value of k for which the function f(x) = { x^2 + k, x < 1; 2x + 3, x >= 1 } is continuous at x = 1.

Correct Answer: 4

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. Since x < 1, we use f(x) = x^2 + k. So, f(1) = 1^2 + k = 1 + k.
Step 4: Calculate the value of the second part of the function at x = 1. Since x >= 1, we use f(x) = 2x + 3. So, f(1) = 2(1) + 3 = 2 + 3 = 5.
Step 5: Set the two results equal to each other to ensure continuity: 1 + k = 5.
Step 6: Solve for k by subtracting 1 from both sides: k = 5 - 1.
Step 7: Simplify the equation to find k: k = 4.

Continuity of Piecewise Functions – Understanding how to ensure that a piecewise function is continuous at a given point by equating the limits from both sides.
Limit Evaluation – Evaluating the limits of the function as it approaches the point of interest from both sides.
Algebraic Manipulation – Solving equations to find unknown constants that ensure continuity.