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

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

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

Step 1: Identify the function f(x) which has two parts: f(x) = x^3 - 3x + b for x < 1 and f(x) = 2x + 1 for x >= 1.
Step 2: To find the value of b that makes the function continuous at x = 1, we need to ensure that the two parts of the function equal each other at x = 1.
Step 3: Calculate the value of the first part of the function at x = 1: f(1) = 1^3 - 3(1) + b = 1 - 3 + b = b - 2.
Step 4: Calculate the value of the second part of the function at x = 1: f(1) = 2(1) + 1 = 2 + 1 = 3.
Step 5: Set the two results equal to each other for continuity: b - 2 = 3.
Step 6: Solve for b: b - 2 = 3 means b = 3 + 2, which gives b = 5.

Piecewise Functions – Understanding how to evaluate and ensure continuity at a point for functions defined in pieces.
Continuity at a Point – The requirement that the left-hand limit, right-hand limit, and the function value at that point must all be equal.