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

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Question: For which value of k is the function f(x) = { kx + 1, x < 2; 3, x ≥ 2 } continuous at x = 2? (2019)

Options:

Correct Answer: 2

Exam Year: 2019

Solution:

To be continuous at x = 2, k(2) + 1 must equal 3. Thus, 2k + 1 = 3, giving k = 1.

For which value of k is the function f(x) = { kx + 1, x < 2; 3, x ≥ 2 } continuous at x = 2? (2019)

For which value of k is the function f(x) = { kx + 1, x < 2; 3, x ≥ 2 } continuous at x = 2? (2019)

Step 1: Understand that the function f(x) has two parts: one for x less than 2 (kx + 1) and one for x greater than or equal to 2 (3).
Step 2: To find the value of k that makes the function continuous at x = 2, we need to ensure that the two parts of the function meet at that point.
Step 3: Calculate the value of the first part of the function at x = 2. This means substituting x with 2 in the first part: f(2) = k(2) + 1.
Step 4: Set the result from Step 3 equal to the value of the second part of the function at x = 2, which is 3. So, we have the equation: 2k + 1 = 3.
Step 5: Solve the equation from Step 4 for k. First, subtract 1 from both sides: 2k = 3 - 1, which simplifies to 2k = 2.
Step 6: Divide both sides of the equation by 2 to find k: k = 2 / 2, which gives k = 1.

Continuity of Piecewise Functions – Understanding how to determine the continuity of a function defined in pieces by ensuring the limits from both sides match at a given point.
Solving Linear Equations – Applying algebraic methods to solve for the variable k in the equation derived from the continuity condition.