Find the value of b for which the function f(x) = { x^2 + b, x < 1; 2x + 3, x

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Question: Find the value of b for which the function f(x) = { x^2 + b, x < 1; 2x + 3, x >= 1 is continuous at x = 1.

Options:

Correct Answer: 2

Solution:

Setting 1 + b = 2 + 3 gives b = 4.

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

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

Step 1: Understand that we want the function f(x) to be continuous at x = 1.
Step 2: For f(x) to be continuous at x = 1, the value of f(x) when approaching from the left (x < 1) must equal the value of f(x) when approaching from the right (x >= 1).
Step 3: Calculate f(1) using the right side of the function: f(1) = 2(1) + 3 = 2 + 3 = 5.
Step 4: Calculate the left limit as x approaches 1 using the left side of the function: f(1) = 1^2 + b = 1 + b.
Step 5: Set the left limit equal to the right limit: 1 + b = 5.
Step 6: Solve for b: b = 5 - 1 = 4.

Continuity of Piecewise Functions – The question tests the understanding of how to ensure continuity at a point for piecewise functions by equating the limits from both sides.