If f(x) = { x^2, x < 3; k, x = 3; 2x, x > 3 } is continuous at x = 3, what

If f(x) = { x^2, x < 3; k, x = 3; 2x, x > 3 } is continuous at x = 3, what is the value of k?

If f(x) = { x^2, x < 3; k, x = 3; 2x, x > 3 } is continuous at x = 3, what is the value of k?

Step 1: Understand that the function f(x) has different expressions depending on the value of x.
Step 2: Identify the three parts of the function: f(x) = x^2 when x < 3, f(x) = k when x = 3, and f(x) = 2x when x > 3.
Step 3: For the function to be continuous at x = 3, the limit of f(x) as x approaches 3 from the left must equal f(3).
Step 4: Calculate the limit of f(x) as x approaches 3 from the left: since x < 3, use f(x) = x^2. So, limit as x approaches 3 from the left is 3^2 = 9.
Step 5: Set the limit equal to f(3): 9 = k.
Step 6: Solve for k: k = 9.

Continuity of Functions – Understanding the conditions for a function to be continuous at a point, specifically that the limit from the left must equal the limit from the right and the function's value at that point.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value, and ensuring the transitions between these expressions maintain continuity.
Limits – Calculating the limit of a function as it approaches a specific point from both sides.