For the function f(x) = { x^2, x < 3; 9, x = 3; 3x, x > 3 } to be continuo

For the function f(x) = { x^2, x < 3; 9, x = 3; 3x, x > 3 } to be continuous at x = 3, the value of f(3) must be:

For the function f(x) = { x^2, x < 3; 9, x = 3; 3x, x > 3 } to be continuous at x = 3, the value of f(3) must be:

Step 1: Understand that for a function to be continuous at a point, the value of the function at that point must equal the limit of the function as it approaches that point.
Step 2: Identify the point we are checking for continuity, which is x = 3 in this case.
Step 3: Look at the definition of the function f(x) at x = 3. It states that f(3) = 9.
Step 4: Calculate the limit of f(x) as x approaches 3 from the left (x < 3). This is f(x) = x^2, so as x approaches 3, f(x) approaches 3^2 = 9.
Step 5: Calculate the limit of f(x) as x approaches 3 from the right (x > 3). This is f(x) = 3x, so as x approaches 3, f(x) approaches 3*3 = 9.
Step 6: Since both the left-hand limit and the right-hand limit as x approaches 3 equal 9, the overall limit as x approaches 3 is also 9.
Step 7: For the function to be continuous at x = 3, we need f(3) to equal this limit, which is 9.

Continuity of Functions – A function is continuous at a point if the limit as x approaches that point equals the function's value at that point.
Piecewise Functions – Understanding how to evaluate limits and function values in piecewise-defined functions.