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

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

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

Correct Answer: 0

Step 1: Understand that we have a piecewise function f(x) which has two parts: one for x < 3 and another for x >= 3.
Step 2: Identify the two parts of the function: f(x) = 2x + m for x < 3 and f(x) = x^2 - 3 for x >= 3.
Step 3: To find the value of m that makes the function continuous at x = 3, we need to ensure that both parts of the function give the same output when x = 3.
Step 4: Calculate the value of the second part of the function at x = 3: f(3) = 3^2 - 3 = 9 - 3 = 6.
Step 5: Now, set the first part of the function equal to this value when x approaches 3 from the left: 2(3) + m = 6.
Step 6: Simplify the equation: 6 + m = 6.
Step 7: Solve for m by subtracting 6 from both sides: m = 6 - 6 = 0.

Piecewise Functions – Understanding how to evaluate and ensure continuity at a point where a piecewise function changes its definition.
Continuity – The condition that the left-hand limit and right-hand limit at a point must equal the function's value at that point.