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Floor and Ceiling Functions

Floor functions map a real number x to the largest integer less than or equal to x. You can probably guess what the ceiling function does.

Floor Function

         

What is \(\left\lfloor\sqrt{46}\right\rfloor?\)

Which of the following statements is/are true?

I. \(\lfloor x+n\rfloor=\lfloor x\rfloor+n\) for any real \(x\) and any integer \(n.\)
II. \(\lfloor x\rfloor+\lfloor-x\rfloor=0\) if and only if \(x\) is an integer.
III. \(\lfloor x\rfloor+\lfloor y\rfloor\le\lfloor x+y\rfloor\le\lfloor x\rfloor+\lfloor y\rfloor+1\) for any real \(x\) and \(y.\)

What is the solution set to \(\left\lfloor \frac{x}{2}\right\rfloor=4?\)

How many integers \(x\) satisfy the equation \[\lfloor\sqrt{x}\rfloor=3?\]

What is \(\lfloor \pi \rfloor?\)

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