To determine which statements are true about the function f(x), analyze the function in different regions. The correct answer is D. I and II only.
Step-by-step explanation:
To determine which of the given statements are true about the function f(x), we need to analyze the function in different regions.
In region I (x ≤ 2), the function is defined as f(x) = x - 3. In region II (x > 2), the function is defined as f(x) = 2x + 1.
Now, let's evaluate each statement:
f(2) exists: Since 2 is included in the domain of region I, we can substitute 2 into the function f(x) = x - 3 and get f(2) = 2 - 3 = -1. So, this statement is true.
f is continuous at 2: Since the function has different definitions for x ≤ 2 and x > 2, the function is not continuous at x = 2. Therefore, this statement is false.
lim f(x) exists: The limit of f(x) as x approaches 2 from the left side (region I) is x - 3, and the limit as x approaches 2 from the right side (region II) is 2x + 1. Since both limits approach the same value (-1), the limit of f(x) as x approaches 2 exists. So, this statement is true.
Based on the above analysis, the correct answer is D. I and II only.