Find the value of a for which the function f(x) = { ax + 1, x < 1; 3, x = 1;

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

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

Step 1: Understand that we need to find the value of 'a' so that the function f(x) is continuous at x = 1.
Step 2: Recall that for a function to be continuous at a point, the left-hand limit and right-hand limit at that point must equal the function's value at that point.
Step 3: Identify the function's value at x = 1, which is f(1) = 3.
Step 4: For x < 1, the function is f(x) = ax + 1. We need to find the limit as x approaches 1 from the left: f(1-) = a(1) + 1 = a + 1.
Step 5: Set the left-hand limit equal to the function's value at x = 1: a + 1 = 3.
Step 6: Solve for 'a': a + 1 = 3 leads to a = 3 - 1, so a = 2.
Step 7: For x > 1, the function is f(x) = 2x + a. We need to find the limit as x approaches 1 from the right: f(1+) = 2(1) + a = 2 + a.
Step 8: Set the right-hand limit equal to the function's value at x = 1: 2 + a = 3.
Step 9: Solve for 'a': 2 + a = 3 leads to a = 3 - 2, so a = 1.
Step 10: Since both limits must equal the function value at x = 1, we confirm that a = 2 satisfies both conditions.

Continuity of Piecewise Functions – The question tests the understanding of how to ensure a piecewise function is continuous at a specific point by equating the function values from different intervals.
Limit and Function Value Matching – It requires the student to set the left-hand limit and right-hand limit equal to the function value at the point of interest to find the unknown parameter.