For which value of p is the function f(x) = { x^2 + p, x < 0; 3x - 1, x >=

For which value of p is the function f(x) = { x^2 + p, x < 0; 3x - 1, x >= 0 } continuous at x = 0?

For which value of p is the function f(x) = { x^2 + p, x < 0; 3x - 1, x >= 0 } continuous at x = 0?

Correct Answer: -1

Step 1: Identify the two pieces of the function f(x). The function is defined as f(x) = x^2 + p for x < 0 and f(x) = 3x - 1 for x >= 0.
Step 2: Find the value of f(0) using the second piece of the function since 0 is included in that piece. Calculate f(0) = 3(0) - 1 = -1.
Step 3: To ensure the function is continuous at x = 0, the limit of f(x) as x approaches 0 from the left (x < 0) must equal f(0).
Step 4: Calculate the limit of f(x) as x approaches 0 from the left. This means using the first piece: limit as x approaches 0 from the left of f(x) = x^2 + p = 0^2 + p = p.
Step 5: Set the limit from the left equal to f(0): p = -1.
Step 6: Conclude that for the function to be continuous at x = 0, p must equal -1.

Piecewise Functions – Understanding how to evaluate and ensure continuity at a point where a piecewise function changes its definition.
Continuity at a Point – The requirement that the left-hand limit, right-hand limit, and the function value at that point must all be equal.