Greatest Integer Functions
Date: 09/27/98 at 18:10:45 From: Laura Subject: Greatest integer function I am in 11th grade pre-calculus. We have been asked to solve for the graph of [y]=[x]. I am familiar with the greatest integer function of y=[x], but I do not know how to go about solving [y]=[x]. Please help. Thank you, Laura
Date: 09/27/98 at 18:30:56 From: Doctor Pat Subject: Re: Greatest integer function Laura, This is an unusual question, but interesting. Let's see if we can figure out what it means. We want to mark all the points on the coordinate plane that would make this equation true. First it should be obvious that if y = x then [y] = [x] is true so the line y = x must be part of the solution. Okay? Now, think about numbers that give the same result when using [x]. Any number x so that 0 <= x < 1 has a value of 0. That would be true for 0 <= y < 1 also. So any ordered pair (x,y) where both are in the interval [0,1), would be included in our solution. Following this idea for values between [1,2) etc. it seems that we should expect to have a series of squares moving along the line y = x. Shade the square where x and y are both between zero and one. The x-axis and y-axis, including the origin, are part of the solution out to x = 1 and y = 1, which are not in the solution. Another square starts at (1,1) and goes to (1,2), (2,2), (2,1). Again, the bottom and left edge of the square are in the solution and the top and right edge of each square are not in the solution. Continue to infinity in both directions. If you could see all the arm waving I'm doing it might help, but alas, that is not yet possible. Good luck, - Doctor Pat, The Math Forum http://mathforum.org/dr.math/
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