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A floor function equation M421 in CRUX

Shenghui Yang, WOLFRAM
Posted 3 years ago

POSTED BY: Shenghui Yang
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Hello, could you help me with my question? I have this function and I know that the range is a union of several intervals. Is there any way for Mathematica to give me the general formula of the range of this function?

(Abs[x] - Floor[x])/(Sqrt[2 - Floor[x]])

I found the ranges by defining an interval for x, following the example of what you wrote. 

Reduce[FunctionRange[{(Abs[x] - Floor[x])/Sqrt[2 - Floor[x]], -50 <= 
    x < 0}, x, y]]

This is the plot:

Plot[(Abs[x] - Floor[x])/(Sqrt[2 - Floor[x]]), {x, -20, 3}]

enter image description here
POSTED BY: Emerson Alex Villafuerte Jara

Three things to consider:

  1. Only consider one continuous subdomain for x at a time. Floor function introduces non-linearity to the solution and in general no silver bullet to kill all problems.
  2. Beware of floor function for negative numbers. Test some numbers if you find it hard to memorize the behavior
  3. Plot on small scale and use customized grid lines to visualize the boundaries for jumpy ranges.

Only discuss the negative integer cases here:

res

Of course you also need to add the range for the positive part, which is easy to handle by choosing x in [0,1) and [1,2), or k = 0 and 1.
POSTED BY: Shenghui Yang

Thank you very much, sir
POSTED BY: Emerson Alex Villafuerte Jara

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