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Define \(\{x\} = x-\lfloor x \rfloor\). That is to say, \(\{x\}\) is the "fractional part" of \(x\). For example, if you were to expand a positive number  as a decimal, \(\{x\}\) is the stuff after the decimal point. For example \(\left\{\frac{3}{2}\right\} = 0.5\) and \(\{\pi\} = 0.14159\dots\)

Now, using the above definition, determine if the function below is increasing, decreasing, even, odd, and/or invertible on its natural domain:


\(f(x) = \lfloor x \rfloor - \left\{ x \right\}\)

 

For each property, write inCreasing, Decreasing, Even, Odd, inVertible in that order (alphabetical).

 Jun 8, 2022
 #1
avatar+117487 
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I'm finding the notation really confusing so I am going to change it

 

\(x=Z+\alpha \qquad \text{where } Z=\lfloor x \rfloor\;\; and \;\;\;0\le\alpha<1\)

 

\(f(x)=f(Z+\alpha)=Z-\alpha\)

 

f(3.2)= 3-0.2=2.8

f(-3.2)=f(-4+0.8) =-4-0.8 = -4.8

 

So this function certainly isn't even of odd.

It is not continuous either.

The continuous segments are decreasing but I doubt that is valid, the graph as a whole is not decreasing.

 

anyway, that is something for you to think about.

 

Here is the graph

 

 Jun 8, 2022

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