What is the value of k for which the function f(x) = { kx + 2, x < 2; x^2 - 4
Practice Questions
Q1
What is the value of k for which the function f(x) = { kx + 2, x < 2; x^2 - 4, x >= 2 is continuous at x = 2?
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Questions & Step-by-Step Solutions
What is the value of k for which the function f(x) = { kx + 2, x < 2; x^2 - 4, x >= 2 is continuous at x = 2?
Step 1: Understand that we want the function f(x) to be continuous at x = 2.
Step 2: Identify the two parts of the function: for x < 2, f(x) = kx + 2, and for x >= 2, f(x) = x^2 - 4.
Step 3: Find the value of f(2) using the second part of the function since 2 is included in x >= 2. Calculate f(2) = 2^2 - 4 = 0.
Step 4: Find the limit of f(x) as x approaches 2 from the left (x < 2). This means using the first part of the function: limit as x approaches 2 from the left is k(2) + 2 = 2k + 2.
Step 5: Set the limit from the left equal to the value of the function at x = 2 for continuity: 2k + 2 = 0.
Step 6: Solve the equation 2k + 2 = 0. Subtract 2 from both sides: 2k = -2.
