Find the value of k such that the function f(x) = { kx, x < 0; x^2 + 1, x >

Find the value of k such that the function f(x) = { kx, x < 0; x^2 + 1, x >= 0 is continuous at x = 0.

Find the value of k such that the function f(x) = { kx, x < 0; x^2 + 1, x >= 0 is continuous at x = 0.

Step 1: Understand that we need to find the value of k so that the function f(x) is continuous at x = 0.
Step 2: Recall that for a function to be continuous at a point, the left-hand limit and the right-hand limit at that point must be equal to the function's value at that point.
Step 3: Identify the two parts of the function: f(x) = kx for x < 0 and f(x) = x^2 + 1 for x >= 0.
Step 4: Calculate the left-hand limit as x approaches 0 from the left (x < 0). This is f(0) = k(0) = 0.
Step 5: Calculate the right-hand limit as x approaches 0 from the right (x >= 0). This is f(0) = 0^2 + 1 = 1.
Step 6: Set the left-hand limit equal to the right-hand limit: 0 = 1.
Step 7: Since we need the left-hand limit (0) to equal the right-hand limit (1), we set k(0) equal to 1: k(0) = 1.
Step 8: Solve for k: Since k(0) = 1, we find that k must be 1.

Continuity of Functions – Understanding the conditions under which a piecewise function is continuous at a point, specifically ensuring that the left-hand limit equals the right-hand limit and the function value at that point.