Question

Let f(x)=

x-3 if x ≤2
2x+1 if x > 2
4)
Which of the following statements are true about f?
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A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
1. f(2) exists
II. f is continuous at 2
III. lim f(x) exists
x-2

Answer 1

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.