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

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

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

Step 1: Identify the function f(x) which has different expressions based on the value of x: f(x) = 3x + m for x < 1, f(x) = 2 for x = 1, and f(x) = mx + 1 for x > 1.
Step 2: To ensure the function is continuous at x = 1, the value of f(x) as x approaches 1 from the left (3x + m) must equal the value at x = 1 (which is 2).
Step 3: Set up the equation from the left side: 3(1) + m = 2. This simplifies to 3 + m = 2.
Step 4: Solve for m: m = 2 - 3, which gives m = -1.
Step 5: Next, ensure the value of f(x) as x approaches 1 from the right (mx + 1) also equals 2. Set up the equation: m(1) + 1 = 2.
Step 6: Solve for m: m + 1 = 2, which simplifies to m = 2 - 1, giving m = 1.
Step 7: The value of m must satisfy both conditions. The only value that works for both is m = 1.

Piecewise Functions – Understanding how to evaluate and ensure continuity in piecewise-defined functions.
Continuity at a Point – Applying the definition of continuity to find values that make a function continuous at a specific point.
Solving Equations – Setting up and solving equations derived from the conditions for continuity.