Find the value of k such that the function f(x) = { kx + 1, x < 1; 3, x = 1;

Find the value of k such that the function f(x) = { kx + 1, x < 1; 3, x = 1; x^2 + 1, x > 1 is continuous at x = 1.

Find the value of k such that the function f(x) = { kx + 1, x < 1; 3, x = 1; x^2 + 1, x > 1 is continuous at x = 1.

Step 1: Understand that the function f(x) has different expressions based on the value of x. We need to check the value of f(x) at x = 1.
Step 2: Identify the value of f(x) when x = 1. According to the function, f(1) = 3.
Step 3: For the function to be continuous at x = 1, the limit of f(x) as x approaches 1 from the left (x < 1) must equal f(1).
Step 4: The expression for f(x) when x < 1 is kx + 1. We need to find the limit as x approaches 1 from the left: limit as x approaches 1 of (kx + 1) = k(1) + 1 = k + 1.
Step 5: Set the limit equal to f(1): k + 1 = 3.
Step 6: Solve for k: k + 1 = 3 means k = 3 - 1, which simplifies to k = 2.

Continuity of Piecewise Functions – Understanding how to ensure a piecewise function is continuous at a specific point by matching the limits from both sides.
Limit Evaluation – Evaluating the left-hand limit and right-hand limit at the point of interest to find the necessary conditions for continuity.