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

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

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

Step 1: Understand that we want the function f(x) to be continuous at x = 1.
Step 2: Recall that for a function to be continuous at a point, the left-hand limit and the right-hand limit at that point must be equal to the function's value at that point.
Step 3: Identify the function's definition: f(x) = ax + 2 for x < 1 and f(x) = 3 for x >= 1.
Step 4: Since we are looking at x = 1, we need to find the left-hand limit as x approaches 1 from the left (x < 1). This is given by ax + 2.
Step 5: Substitute x = 1 into the left-hand function: f(1) = a(1) + 2 = a + 2.
Step 6: The right-hand limit as x approaches 1 from the right (x >= 1) is simply f(1) = 3.
Step 7: Set the left-hand limit equal to the right-hand limit: a + 2 = 3.
Step 8: Solve for a: a + 2 = 3 leads to a = 3 - 2, which simplifies to 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 equals the right-hand limit.