For which value of a is the function f(x) = { ax + 1, x < 0; 2, x = 0; 3x - 1

For which value of a is the function f(x) = { ax + 1, x < 0; 2, x = 0; 3x - 1, x > 0 } continuous at x = 0?

For which value of a is the function f(x) = { ax + 1, x < 0; 2, x = 0; 3x - 1, x > 0 } continuous at x = 0?

Correct Answer: 2

Step 1: Understand that the function f(x) has different expressions depending on the value of x: ax + 1 for x < 0, 2 for x = 0, and 3x - 1 for x > 0.
Step 2: To find the value of a that makes the function continuous at x = 0, we need to ensure that the left-hand limit (as x approaches 0 from the left) equals the value of the function at x = 0.
Step 3: The left-hand limit as x approaches 0 from the left is given by the expression ax + 1. We substitute x = 0 into this expression: a(0) + 1 = 1.
Step 4: The value of the function at x = 0 is given as 2.
Step 5: For the function to be continuous at x = 0, we need the left-hand limit (1) to equal the value of the function at x = 0 (2). So we set 1 equal to 2: ax + 1 = 2.
Step 6: Since we are looking for the value of a when x = 0, we can rewrite the equation as 1 = 2, which is not correct. Instead, we need to set ax + 1 equal to 2 when x is approaching 0 from the left.
Step 7: Therefore, we set ax + 1 = 2 and solve for a when x = 0: 1 = 2, which leads us to find that a must equal 2 for the function to be continuous.

Continuity of Piecewise Functions – The question tests the understanding of how to determine the continuity of a piecewise function at a specific point by ensuring the left-hand limit, right-hand limit, and the function value at that point are equal.