For which value of a is the function f(x) = { ax + 1, x < 0; 2x + a, x >=

For which value of a is the function f(x) = { ax + 1, x < 0; 2x + a, x >= 0 } continuous at x = 0?

For which value of a is the function f(x) = { ax + 1, x < 0; 2x + a, x >= 0 } continuous at x = 0?

Correct Answer: 1

Step 1: Identify the function f(x) which has two parts: ax + 1 for x < 0 and 2x + a for x >= 0.
Step 2: To find the value of a that makes the function continuous at x = 0, we need to ensure that both parts of the function give the same output when x = 0.
Step 3: Calculate the output of the first part (ax + 1) when x = 0. This gives us f(0) = a(0) + 1 = 1.
Step 4: Calculate the output of the second part (2x + a) when x = 0. This gives us f(0) = 2(0) + a = a.
Step 5: Set the two outputs equal to each other: 1 = a.
Step 6: Solve for a. This gives us a = 1.

Piecewise Functions – Understanding how to evaluate and ensure continuity at a point for functions defined in pieces.
Continuity – The concept that a function is continuous at a point if the limit from both sides equals the function's value at that point.