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

₹0.0

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

Options:

Correct Answer: 2

Solution:

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

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

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

Step 1: Understand that for a function to be continuous at a point, the value of the function from the left side must equal the value from the right side at that point.
Step 2: Identify the point we are checking for continuity, which is x = 2.
Step 3: Look at the function definition: f(x) = kx + 1 for x < 2 and f(x) = 3 for x >= 2.
Step 4: Calculate the left-hand limit as x approaches 2. This means we use the part of the function for x < 2: f(2) = k(2) + 1.
Step 5: Calculate the right-hand limit as x approaches 2. This means we use the part of the function for x >= 2: f(2) = 3.
Step 6: Set the left-hand limit equal to the right-hand limit: 2k + 1 = 3.
Step 7: Solve the equation 2k + 1 = 3 for k.
Step 8: Subtract 1 from both sides: 2k = 2.
Step 9: Divide both sides by 2: k = 1.

Continuity of Functions – Understanding the conditions under which a piecewise function is continuous at a given point.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value.
Solving Equations – Setting up and solving equations to find unknown parameters in functions.