Find the value of a for which the function f(x) = { x^2 + a, x < 1; 3, x = 1;

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

Find the value of a for which the function f(x) = { x^2 + a, x < 1; 3, x = 1; 2x + 1, 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: Identify the three parts of the function: f(x) = x^2 + a for x < 1, f(x) = 3 for x = 1, and f(x) = 2x + 1 for x > 1.
Step 3: Find the left limit as x approaches 1 from the left (x < 1). This is f(1) = 1^2 + a = 1 + a.
Step 4: Find the right limit as x approaches 1 from the right (x > 1). This is f(1) = 2(1) + 1 = 2 + 1 = 3.
Step 5: Set the left limit equal to the right limit: 1 + a = 3.
Step 6: Solve for 'a': Subtract 1 from both sides to get a = 3 - 1, which simplifies to a = 2.

Continuity of Functions – The question tests the understanding of continuity at a point, specifically how to ensure that the left-hand limit, right-hand limit, and the function value at that point are equal.
Piecewise Functions – The function is defined in pieces, requiring the student to analyze different cases based on the value of x.
Limit Evaluation – The question involves evaluating limits from both sides of a point to determine continuity.