Final answer:
The statement ['x²] = ['x']² is disproven with a simple counterexample using the number 1.5. Squaring 1.5 results in 2.25, where the floor value is 2, whereas the floor value of 1.5 is 1, and its square is 1, showing that the two values are not equal.
Step-by-step explanation:
The student's question whether for all real numbers x, the greatest integer function of x squared [x²] is equal to the greatest integer function of x [x] squared can be approached by looking for a counterexample. We do not need to complete the square or use any complex algebra to find a counterexample. Let's consider a real number that is not an integer, for example, x = 1.5. If we square it, we get x² = 2.25. The greatest integer less than or equal to 2.25 is 2, so [x²] = 2. On the other hand, the greatest integer less than or equal to 1.5 is 1, so [x] = 1, and therefore [x]² = 1. This shows that [x²] ≠ [x]², proving that the initial statement is false.
It is essential to remember that the greatest integer function, also known as the floor function, always rounds down to the nearest integer. In summary, a counterexample is sufficient to demonstrate that the original statement does not hold for all real numbers.