For which value of b is the function f(x) = { 2x + 1, x < 1; b, x = 1; x^2 +

For which value of b is the function f(x) = { 2x + 1, x < 1; b, x = 1; x^2 + 1, x > 1 continuous at x = 1?

For which value of b is the function f(x) = { 2x + 1, x < 1; b, x = 1; x^2 + 1, x > 1 continuous at x = 1?

Step 1: Understand that we want the function f(x) to be continuous at x = 1.
Step 2: Recall that for a function to be continuous at a point, the left limit, right limit, and the function value at that point must all be equal.
Step 3: Identify the left limit as x approaches 1 from the left (x < 1). This is given by the function 2x + 1.
Step 4: Calculate the left limit: 2(1) + 1 = 2 + 1 = 3.
Step 5: Identify the right limit as x approaches 1 from the right (x > 1). This is given by the function x^2 + 1.
Step 6: Calculate the right limit: (1)^2 + 1 = 1 + 1 = 2.
Step 7: Set the left limit equal to the right limit to find the value of b: 3 = b.
Step 8: Conclude that for the function to be continuous at x = 1, b must equal 3.

Continuity of Piecewise Functions – Understanding how to determine the continuity of a piecewise function at a specific point by evaluating limits from both sides and ensuring they equal the function's value at that point.
Limit Evaluation – Calculating left-hand and right-hand limits to find the necessary conditions for continuity.