For which value of a is the function f(x) = { 3x + a, x < 2; 4x - 1, x >=

For which value of a is the function f(x) = { 3x + a, x < 2; 4x - 1, x >= 2 continuous at x = 2?

For which value of a is the function f(x) = { 3x + a, x < 2; 4x - 1, x >= 2 continuous at x = 2?

Step 1: Identify the function f(x) which has two parts: 3x + a for x < 2 and 4x - 1 for x >= 2.
Step 2: To find the value of a that makes the function continuous at x = 2, we need to ensure that the two parts of the function equal each other at x = 2.
Step 3: Calculate the value of the first part of the function at x = 2: f(2) = 3(2) + a.
Step 4: Calculate the value of the second part of the function at x = 2: f(2) = 4(2) - 1.
Step 5: Set the two expressions equal to each other: 3(2) + a = 4(2) - 1.
Step 6: Simplify the equation: 6 + a = 8 - 1.
Step 7: Further simplify: 6 + a = 7.
Step 8: Solve for a by subtracting 6 from both sides: a = 7 - 6.
Step 9: Conclude that a = 1.

Continuity of Piecewise Functions – The question tests the understanding of how to ensure continuity at a point for piecewise functions by equating the limits from both sides.
Limit Evaluation – It requires evaluating the limits of the function as x approaches the point of interest (x = 2) from both sides.