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

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

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

Step 1: Understand that we need to find the value of m that makes the function f(x) 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: f(x) = 2x + m for x < 1 and f(x) = mx + 3 for x >= 1.
Step 4: Calculate the left-hand limit as x approaches 1: f(1) from the left is 2(1) + m = 2 + m.
Step 5: Calculate the right-hand limit as x approaches 1: f(1) from the right is m(1) + 3 = m + 3.
Step 6: Set the left-hand limit equal to the right-hand limit: 2 + m = m + 3.
Step 7: Solve the equation: Subtract m from both sides to get 2 = 3, which simplifies to m = 1.
Step 8: Conclude that the value of m for which the function is continuous at x = 1 is 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 analyze both parts of the function to find the value of m.