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

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

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

Correct Answer: -1

Step 1: Understand that we need to find the value of m 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 the right-hand limit at that point must be equal to the function's value at that point.
Step 3: Identify the two parts of the function: for x < 1, f(x) = 3x + m, and for x >= 1, f(x) = 2x^2.
Step 4: Calculate the left-hand limit as x approaches 1 from the left (x < 1): f(1) = 3(1) + m = 3 + m.
Step 5: Calculate the right-hand limit as x approaches 1 from the right (x >= 1): f(1) = 2(1)^2 = 2.
Step 6: Set the left-hand limit equal to the right-hand limit: 3 + m = 2.
Step 7: Solve for m: m = 2 - 3, which gives m = -1.

Continuity of Functions – The question tests the understanding of continuity at a point, specifically how to ensure that the left-hand limit and right-hand limit at x = 1 are equal.
Piecewise Functions – The function is defined in pieces, requiring the student to evaluate the function differently based on the value of x.
Limit Evaluation – The student must evaluate the limits from both sides of x = 1 to find the value of m that makes the function continuous.