+166 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).
+118673 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
