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

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Question: For which value of k is the function f(x) = { kx + 1, x < 2; 3, x = 2; 2x - 1, x > 2 } continuous at x = 2?

Options:

Correct Answer: 2

Solution:

To ensure continuity at x = 2, k(2) + 1 must equal 3. Thus, k = 1.

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

For which value of k is the function f(x) = { kx + 1, x < 2; 3, x = 2; 2x - 1, x > 2 } 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 value at x = 2, which is given as 3.
Step 3: Determine the left-hand limit as x approaches 2. This is found using the expression for x < 2, which is kx + 1. So, we calculate k(2) + 1.
Step 4: Set the left-hand limit equal to the function value at x = 2. This gives us the equation k(2) + 1 = 3.
Step 5: Solve the equation from Step 4. First, simplify it: 2k + 1 = 3.
Step 6: Subtract 1 from both sides: 2k = 2.
Step 7: Divide both sides by 2 to find k: k = 1.
Step 8: Conclude that the value of k that makes the function continuous at x = 2 is 1.

Continuity of Functions – The concept of continuity at a point requires that the left-hand limit, right-hand limit, and the function's value at that point are all equal.
Piecewise Functions – Understanding how to evaluate and analyze functions defined by different expressions based on the input value.
Limit Evaluation – The process of finding the limit of a function as it approaches a certain point from both sides.