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Question Number 205297 by mnjuly1970 last updated on 15/Mar/24
Find the′′ range ′′ of : i : f (x) =⌊ (( x)/( ⌊ x ⌋)) ⌋ ii: f(x) = (( x)/(⌊ x ⌋ + ⌊ −x ⌋))
Findthe″range″of:i:f(x)=⌊x⌊x⌋⌋ii:f(x)=x⌊x⌋+⌊−x⌋
Answered by A5T last updated on 15/Mar/24
(i)f is undefined when ⌊x⌋=0 (0≤x<1) Otherwise x=⌊x⌋+{x},where 0≤{x}<1 ⇒f(x)=⌊1+(({x})/(⌊x⌋))⌋, when x≥1,{x}<⌊x⌋,⇒f(x)=1 when x<0 f(x)=⌊1−(({x})/(∣⌊x⌋∣))⌋; {x}<∣⌊x⌋∣ ⇒0<1−(({x})/(∣⌊x⌋∣))≤1⇒ f(x)_(x<0) = 0 or 1 ⇒Range: {0,1}
(i)fisundefinedwhen⌊x⌋=0(0⩽x<1)Otherwisex=⌊x⌋+{x},where0⩽{x}<1⇒f(x)=⌊1+{x}⌊x⌋⌋,whenx⩾1,{x}<⌊x⌋,⇒f(x)=1whenx<0f(x)=⌊1−{x}∣⌊x⌋∣⌋;{x}<∣⌊x⌋∣⇒0<1−{x}∣⌊x⌋∣⩽1⇒f(x)x<0=0or1⇒Range:{0,1}
Commented by mnjuly1970 last updated on 15/Mar/24
thanks a lot ali
thanksalotali
Commented by Frix last updated on 15/Mar/24
Yes. Just to clarify: f(x)= { ((1; x∈Z^− )),((0; x∈R^− \Z)),((undefined; 0≤x<1)),((1; x≥1)) :}
Yes.Justtoclarify:f(x)={1;x∈Z−0;x∈R−∖Zundefined;0⩽x<11;x⩾1
Answered by A5T last updated on 15/Mar/24
(ii) when x∈Z,f(x) is undefined when x>0, ⌊−x⌋=−⌊x⌋−1⇒f(x)=−x when x<0, ⌊−x⌋=−⌊x⌋−1⇒f(x)=−x ⇒f(x)∈R\Z
(ii)whenx∈Z,f(x)isundefinedwhenx>0,⌊−x⌋=−⌊x⌋−1⇒f(x)=−xwhenx<0,⌊−x⌋=−⌊x⌋−1⇒f(x)=−x⇒f(x)∈R∖Z

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