To determine if the function f(x) is continuous at x=3, we need to check three conditions: 1. f(3) is defined. 2. The limit as x approaches 3 of f(x) exists. 3. The limit as x approaches 3 of f(x) equals f(3). Let's check these conditions step by step: 1. f(3) is defined: For x

Question

To determine if the function f(x) is continuous at x=3, we need to check three conditions:

1. f(3) is defined.
2. The limit as x approaches 3 of f(x) exists.
3. The limit as x approaches 3 of f(x) equals f(3).

Let's check these conditions step by step:

1. f(3) is defined:
   For x<=3, f(x) = (x+1)^2. So, f(3) = (3+1)^2 = 16.

2. The limit as x approaches 3 of f(x) exists:
   We need to check the limit from the left and the right side of 3.

From the left side (x<=3), f(x) = (x+1)^2. So, the limit as x approaches 3 from the left is (3+1)^2 = 16.

From the right side (x>3), f(x) = -2x + 22. So, the limit as x approaches 3 from the right is -2*3 + 22 = 16.

Since the limit from the left and the right are equal, the limit as x approaches 3 of f(x) exists and is 16.

3. The limit as x approaches 3 of f(x) equals f(3):
   As we found in the previous steps, both the limit as x approaches 3 of f(x) and f(3) are 16.

Since all three conditions are met, the function f(x) is continuous at x=3. So, the correct answer is A. Continuous at x equals 3.

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