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?
I want to say that you are correct.
Yes, your reasoning is correct. According to the given information, g(x) is continuous for all real numbers. This means that there are no jumps, holes, or asymptotes in the graph of the function.
Let's analyze each statement to see if it necessarily follows from the given information:
I. g(x) = 1 at least once: Since g(x) is continuous for all real numbers and g(3) = -1, g(4) = 2, and there are no jumps or holes in the graph, the function must cross the value of 1 at some point. So, statement I is necessarily true.
II. lim g(x) = g(3.5) as x approaches 3.5: This statement is true because when g(x) is continuous at x = 3.5, the limit of g(x) as x approaches 3.5 will be equal to the value of g(3.5). Therefore, statement II is necessarily true.
III. lim g(x) as x approaches 3 from the left = lim g(x) as x approaches from the right: This statement is true because for a function to be continuous at a point, the limit from the left side must be equal to the limit from the right side. Since g(x) is continuous for all real numbers, this property holds true for any point. Therefore, statement III is necessarily true.
Based on the above analysis, all three statements (I, II, and III) are necessarily true. Hence, the correct answer is D. I, II, and III.