Determine the value of p for which the function f(x) = { 2x + 3, x < 2; px +

Determine the value of p for which the function f(x) = { 2x + 3, x < 2; px + 1, x = 2; x^2 - 1, x > 2 is continuous at x = 2.

Determine the value of p for which the function f(x) = { 2x + 3, x < 2; px + 1, x = 2; x^2 - 1, x > 2 is continuous at x = 2.

Step 1: Identify the function f(x) which is defined in three parts based on the value of x.
Step 2: Recognize that we need to check the continuity of f(x) at x = 2.
Step 3: For continuity at x = 2, the left-hand limit (as x approaches 2 from the left) must equal the value of the function at x = 2, which must also equal the right-hand limit (as x approaches 2 from the right).
Step 4: Calculate the left-hand limit: f(2) from the left is given by the first part of the function, so we calculate 2(2) + 3.
Step 5: Calculate the left-hand limit: 2(2) + 3 = 4 + 3 = 7.
Step 6: The value of the function at x = 2 is given by the second part of the function, which is px + 1. So, we substitute x = 2: f(2) = p(2) + 1 = 2p + 1.
Step 7: Calculate the right-hand limit: f(2) from the right is given by the third part of the function, so we calculate (2)^2 - 1.
Step 8: Calculate the right-hand limit: (2)^2 - 1 = 4 - 1 = 3.
Step 9: Set the left-hand limit equal to the value of the function at x = 2: 7 = 2p + 1.
Step 10: Solve for p: 7 - 1 = 2p, which simplifies to 6 = 2p, and then divide both sides by 2 to find p = 3.

Piecewise Functions – Understanding how to evaluate and ensure continuity in piecewise-defined functions.
Continuity at a Point – Determining the conditions under which a function is continuous at a specific point.
Limit and Function Value Equality – Applying the concept that for continuity, the limit from both sides must equal the function value at that point.