Final answer:
A counterexample to the statement is X = 2.7, where the greatest integer function gives 2, not 3 as would be found if rounded to the nearest integer. Similarly, for X = -1.2, the function yields -2 instead of -1, demonstrating that rounding to the nearest integer is not equivalent to the greatest integer function.
Step-by-step explanation:
To find a counterexample to the statement “in order to find the greatest integer function of X when X is not an integer, round X to the nearest integer,” let's consider the value of X = 2.7. The greatest integer function, also known as the floor function, outputs the largest integer less than or equal to a given number. Therefore, the greatest integer function of 2.7 is 2, not 3 as one might get if they rounded to the nearest integer.
In another instance, if X = -1.2, the greatest integer function would yield -2, because -2 is the largest integer less than -1.2. However, rounding -1.2 to the nearest integer would give -1, clearly showing a discrepancy. This reinforces that the greatest integer function does not round numbers to the nearest integer but rather to the nearest lower integer for non-integer values.