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Posted April 18th, 2010 by gokudeep
in

how is the following function odd function
$ f(x)= (-1)^{[x]} $
where [x] is a greatest integer or floor of x function

‹ functions Permutation & Combination ›

Comments

Odd function

On April 18th, 2010 Structure says:

Let $ x>0 $ and $ [x]\le x<[x]+1 $
Then

$$-[x]-1<-x\le -[x]$$

So for $ x\in (0,\infty)\setminus N $ we have $ [-x]=-[x]-1 $ so $ f(-x)=(-1)^{[-x]}=(-1)^{-1-[x]}=-(-1)^{-[x]}=-(-1)^{[x]}=-f(x) $
Unfortunately for $ n\in N $ $ [n]=n $ and $ [-n]=-n $ so you can not tell that your function is everywhere witg property f(-x)=-f(x).

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