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let-f-x-1-x-1-x-1-prove-that-f-is-periodic-with-period-1-2-give-a-expression-of-f-x-when-x-0-1-

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Question Number 40090 by maxmathsup by imad last updated on 15/Jul/18
let f(x)= 1−[x]−[1−x] 1) prove that f is periodic with period 1 2) give a expression of f(x) when x∈[0,1[
letf(x)=1−[x]−[1−x]1)provethatfisperiodicwithperiod12)giveaexpressionoff(x)whenx∈[0,1[
Commented by math khazana by abdo last updated on 29/Jul/18
1) first we have f(x)=1−[x]−1−[−x] =−[x]−[−x] ⇒ f(x+1) =−[x+1]−[−(x+1)] =−[x]−1−[−x]+1=f(x) so f is periodic with T=1 2) if 0≤x<1 [x]=0 and −1<−x≤0 ⇒[−x]=−1 ⇒ f(x)=−[−x]=−(−1)=1 .
1)firstwehavef(x)=1−[x]−1−[−x]=−[x]−[−x]⇒f(x+1)=−[x+1]−[−(x+1)]=−[x]−1−[−x]+1=f(x)sofisperiodicwithT=12)if0⩽x<1[x]=0and−1<−x⩽0⇒[−x]=−1⇒f(x)=−[−x]=−(−1)=1.

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