If g(x) is continuous for all real numbers and g(3) = -1, g(4) = 2, which of the following are necessarily true?
I. g(x) = 1 at least once
II. lim g(x) = g(3.5) as x aproaches 3.5.
III. lim g(x) as x approaches 3 from the left = lim g(x) as x approaches from the right
A. I only
B. II only
C. I and II only
D. I, II, and III
E. None of these.
I want to say that the answer should be D. I, II, and III since the function is continuous for all real numbers. Would this be correct?
Yes that is correct because if it's continuous then it must pass through 1 and the limit must exist at 3.5 and the limit must equal the same from both sides. So the answer is D (I,II,II)
Well, I must say, you have a good sense of humor! However, let me clarify:
The given information tells us that g(x) is continuous for all real numbers, and we know that continuity implies that a function can be drawn without any breaks or jumps. So, since g(x) is continuous, we can infer the following:
I. Since g(x) is continuous and goes from -1 to 2 (based on g(3) = -1 and g(4) = 2), it must cross the line y = 1 at least once. Therefore, I is necessarily true.
II. The limit of g(x) as x approaches 3.5 should indeed equal to g(3.5) if g(x) is continuous at that point. Since g(x) is continuous for all real numbers, this implies II is also true.
III. The limit of g(x) as x approaches 3 from the left should equal the limit of g(x) as x approaches 3 from the right if g(x) is continuous at x = 3. It is not necessary for g(x) to be continuous at this specific point based on the given information. Therefore, III is not necessarily true.
Hence, the correct answer is C. I and II only. Keep up the humor, but don't take it to the limit of questioning mathematical concepts!
Yes, you are correct. The answer is D. I, II, and III.
Explanation:
If g(x) is continuous for all real numbers, it means that there are no sudden jumps, breaks, or holes in the graph of the function. This means that g(x) can be drawn without lifting the pencil from the paper.
I. g(x) = 1 at least once:
Since the function is continuous, and g(3) = -1 and g(4) = 2, we can conclude that at some point between x = 3 and x = 4, the graph of the function must pass through the value of 1. Therefore, I is necessarily true.
II. lim g(x) = g(3.5) as x approaches 3.5:
Since the function is continuous, we can say that the limit of g(x) as x approaches 3.5 will be equal to the value of g(3.5). Therefore, II is necessarily true.
III. lim g(x) as x approaches 3 from the left = lim g(x) as x approaches 3 from the right:
Since the function is continuous, the limit of g(x) as x approaches 3 from the left will be equal to the limit of g(x) as x approaches 3 from the right. This is because there are no abrupt changes in the behavior of the function around x = 3. Therefore, III is necessarily true.
Hence, the correct answer is D. I, II, and III.
Yes, your answer is correct. The answer is D. I, II, and III. Let me explain why.
To determine the truth of each statement, we need to consider the properties of continuous functions.
Statement I: g(x) = 1 at least once.
Since g(x) is continuous for all real numbers, and it takes the values -1 and 2 at x = 3 and x = 4 respectively, we can infer that the function must pass through every value between -1 and 2 at some point, including 1. Therefore, statement I is true.
Statement II: lim g(x) = g(3.5) as x approaches 3.5.
The second statement deals with the limit of g(x) as x approaches 3.5. According to the continuity of g(x), the limit from both the left and right sides of 3.5 should be equal to g(3.5) for the limit to exist. Since g(3.5) is not given in the information provided, we cannot determine whether this statement is true or false. Therefore, statement II cannot be confirmed based on the given data.
Statement III: lim g(x) as x approaches 3 from the left = lim g(x) as x approaches 3 from the right.
This statement is related to the continuity of g(x). For a function to be continuous at a specific point, the limit from the left side of the point must be equal to the limit from the right side. Since g(x) is continuous for all real numbers, it follows that the limits from the left and right sides of x = 3 are equal. Therefore, statement III is true.
In summary, the correct answer is D. I, II, and III.