For which value of b is the function f(x) = { x^2 - 4, x < 2; bx + 2, x >=

For which value of b is the function f(x) = { x^2 - 4, x < 2; bx + 2, x >= 2 } continuous at x = 2?

For which value of b is the function f(x) = { x^2 - 4, x < 2; bx + 2, x >= 2 } continuous at x = 2?

Correct Answer: -1

Step 1: Identify the two pieces of the function f(x). The first piece is x^2 - 4 for x < 2, and the second piece is bx + 2 for x >= 2.
Step 2: Find the value of the first piece at x = 2. Substitute x = 2 into the first piece: f(2) = 2^2 - 4 = 4 - 4 = 0.
Step 3: Find the value of the second piece at x = 2. Substitute x = 2 into the second piece: f(2) = b(2) + 2 = 2b + 2.
Step 4: Set the two pieces equal to each other at x = 2 to ensure continuity: 0 = 2b + 2.
Step 5: Solve the equation 0 = 2b + 2 for b. Subtract 2 from both sides: -2 = 2b. Then divide both sides by 2: b = -1.

Piecewise Functions – Understanding how to evaluate and ensure continuity at a point where a piecewise function changes its definition.
Continuity – The concept that a function is continuous at a point if the limit from the left equals the limit from the right and equals the function's value at that point.
Solving Equations – The ability to set equations equal to each other and solve for unknown variables.