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.
Practice Questions
1 question
Q1
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.
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Setting k(1) + 1 = 3 gives k = 2 for continuity.
Questions & Step-by-step Solutions
1 item
Q
Q: 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.
Solution: Setting k(1) + 1 = 3 gives k = 2 for continuity.
Steps: 6
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.
