Determine the value of k for which the function f(x) = { x^2 - 4, x < 2; k, x

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Question: Determine the value of k for which the function f(x) = { x^2 - 4, x < 2; k, x = 2; 3x - 4, x > 2 is continuous at x = 2.

Options:

Correct Answer: 2

Solution:

For f(x) to be continuous at x = 2, we need limit as x approaches 2 from left to equal limit as x approaches 2 from right. Thus, k must equal 0.

Determine the value of k for which the function f(x) = { x^2 - 4, x < 2; k, x = 2; 3x - 4, x > 2 is continuous at x = 2.

Determine the value of k for which the function f(x) = { x^2 - 4, x < 2; k, x = 2; 3x - 4, x > 2 is continuous at x = 2.

Step 1: Understand that for a function to be continuous at a point, the left-hand limit, right-hand limit, and the function value at that point must all be equal.
Step 2: Identify the function f(x) and the point of interest, which is x = 2.
Step 3: Calculate the left-hand limit as x approaches 2. This means using the part of the function for x < 2, which is x^2 - 4.
Step 4: Substitute 2 into the left-hand function: (2)^2 - 4 = 4 - 4 = 0.
Step 5: Calculate the right-hand limit as x approaches 2. This means using the part of the function for x > 2, which is 3x - 4.
Step 6: Substitute 2 into the right-hand function: 3(2) - 4 = 6 - 4 = 2.
Step 7: Set the left-hand limit equal to the right-hand limit to find k: 0 (left-hand limit) must equal k (value at x = 2) and also equal 2 (right-hand limit).
Step 8: Since the left-hand limit is 0, for the function to be continuous at x = 2, we must have k = 0.

Continuity of Functions – Understanding the conditions under which a piecewise function is continuous at a specific point.
Limits – Applying the concept of limits to find the value of a function at a point where it is defined piecewise.