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

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

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

Step 1: Understand that we need to find the value of 'a' that makes the function f(x) continuous at x = 0.
Step 2: Recall that a function is continuous at a point if the left-hand limit and right-hand limit at that point are equal to the function's value at that point.
Step 3: Identify the two parts of the function: f(x) = 2x + a for x < 0 and f(x) = x^2 + 1 for x >= 0.
Step 4: Calculate the left-hand limit as x approaches 0 from the left (x < 0): This is f(0-) = 2(0) + a = a.
Step 5: Calculate the right-hand limit as x approaches 0 from the right (x >= 0): This is f(0+) = (0)^2 + 1 = 1.
Step 6: Set the left-hand limit equal to the right-hand limit for continuity: a = 1.
Step 7: Conclude that the value of 'a' that makes the function continuous at x = 0 is a = 1.

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.