Step 1: Identify the function f(x) which is defined in three parts: for x < 2, for x = 2, and for x > 2.
Step 2: Write down the expressions for each part of the function: f(x) = 3x - 1 for x < 2, f(x) = px + 4 for x = 2, and f(x) = x^2 - 2 for x > 2.
Step 3: To find the value of p that makes the function continuous at x = 2, we need to ensure that the left-hand limit (as x approaches 2 from the left) equals the value of the function at x = 2, which equals the right-hand limit (as x approaches 2 from the right).
Step 4: Calculate the left-hand limit: f(2) from the left is 3(2) - 1 = 6 - 1 = 5.
Step 5: The value of the function at x = 2 is f(2) = p(2) + 4 = 2p + 4.
Step 6: Calculate the right-hand limit: f(2) from the right is (2)^2 - 2 = 4 - 2 = 2.
Step 7: Set the left-hand limit equal to the value at x = 2: 5 = 2p + 4.
Step 8: Solve for p: Subtract 4 from both sides to get 1 = 2p, then divide both sides by 2 to find p = 1/2.
Step 9: Check the right-hand limit: Set 2 = 2p + 4 and solve for p to confirm the value is consistent.
